problem solving fraction operations lesson 2.10

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Go Math Grade 6 Answer Key Chapter 2 Fractions and Decimals

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Go Math Grade 6 Chapter 2 Fractions and Decimals Answer Key

Go Math Grade 6 Chapter 2 Fractions and Decimals Solution Key helps you to assess your preparation level. You can easily know which concepts are difficult for preparation and find a simple way to solve the problems using Grade 6 Go Math Answer Key. Learn the concepts easily and apply them to real-life to have a smooth life.

Lesson 1: Fractions and Decimals

• Fractions and Decimals – Page No. 71
• Fractions and Decimals – Page No. 72
• Fractions and Decimals Lesson Check – Page No. 73
• Fractions and Decimals Lesson Check 1 – Page No. 74

Lesson 2: Compare and Order Fractions and Decimals

• Compare and Order Fractions and Decimals – Page No. 77
• Compare and Order Fractions and Decimals – Page No. 78
• Compare and Order Fractions and Decimals Lesson Check – Page No. 79
• Compare and Order Fractions and Decimals Lesson Check 1 – Page No. 80

Lesson 3: Multiply Fractions

• Multiply Fractions – Page No. 83
• Multiply Fractions – Page No. 84
• Multiply Fractions Lesson Check – Page No. 85
• Multiply Fractions Lesson Check 1 – Page No. 86

Lesson 4: Simplify Factors

• Simplify Factors – Page No. 89
• Simplify Factors – Page No. 90
• Simplify Factors Lesson Check – Page No. 91
• Simplify Factors Lesson Check 1 – Page No. 92

Mid-Chapter Checkpoint

• Mid-Chapter Checkpoint – Page No. 93
• Mid-Chapter Checkpoint Lesson Check – Page No. 94

Lesson 5: Investigate • Model Fraction Division

• Model Fraction Division – Page No. 97
• Model Fraction Division – Page No. 98
• Model Fraction Division Lesson Check – Page No. 99
• Model Fraction Division Lesson Check 1 – Page No. 100

Lesson 6: Estimate Quotients

• Estimate Quotients – Page No. 103
• Estimate Quotients – Page No. 104
• Estimate Quotients Lesson Check – Page No. 105
• Estimate Quotients Lesson Check 1 – Page No. 106

Lesson 7: Divide Fractions

• Divide Fractions – Page No. 109
• Divide Fractions – Page No. 110
• Divide Fractions Lesson Check – Page No. 111
• Divide Fractions Lesson Check 1 – Page No. 112

Lesson 8: Investigate • Model Mixed Number Division

• Model Mixed Number Division – Page No. 115
• Model Mixed Number Division – Page No. 116
• Model Mixed Number Division Lesson Check – Page No. 117
• Model Mixed Number Division Lesson Check 1 – Page No. 118

Lesson 9: Divide Mixed Numbers

• Divide Mixed Numbers – Page No. 121
• Divide Mixed Numbers – Page No. 122
• Divide Mixed Numbers Lesson Check – Page No. 123
• Divide Mixed Numbers Lesson Check 1– Page No. 124

Lesson 10: Problem Solving • Fraction Operations

• Fraction Operations – Page No. 127
• Fraction Operations – Page No. 128
• Fraction Operations Lesson Check – Page No. 129
• Fraction Operations Lesson Check 1– Page No. 130

Chapter 2 Review/Test

• Review/Test – Page No. 131
• Review/Test – Page No. 132
• Review/Test – Page No. 133
• Review/Test – Page No. 134
• Review/Test – Page No. 135
• Review/Test – Page No. 136

Share and Show – Page No. 71

Write as a fraction or as a mixed number in simplest form.

Question 1. 95.5 _____ \(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation: 95.5 95.5 is 95 ones and 5 tenths. 5 tenths = \(\frac{5}{10}\) Simplify using the GCF. The GCF of 5 and 10 is 10. Divide the numerator and the denominator by 10 \(\frac{5 ÷ 10}{10 ÷ 10}\) = \(\frac{1}{2}\)

Question 2. 0.6 \(\frac{□}{□}\)

Answer: \(\frac{3}{5}\)

Explanation: 0.6 6 tenths = \(\frac{6}{10}\) Simplify using the GCF. The GCF of 6 and 10 is 2. Divide the numerator and the denominator by 10 \(\frac{6 ÷ 2}{10 ÷ 2}\) = \(\frac{3}{5}\)

Compare Fractions and Decimals Lesson 1 Question 3. 5.75 _____ \(\frac{□}{□}\)

Answer: 5\(\frac{3}{4}\)

Explanation: 5.75 is 5 ones and 75 hundredths. 75 hundredths = \(\frac{75}{100}\) Simplify using the GCF. The GCF of 75 and 100 is 25. Divide the numerator and the denominator by 25 5\(\frac{75 ÷ 25}{100 ÷ 25}\) = 5\(\frac{3}{4}\)

Write as a decimal.

Question 4. \(\frac{7}{8}\) _____

Answer: 0.875

Explanation: Use division to rename the fraction part as a decimal. 7/8 = 0.875 The quotient has 3 decimal places. Add the whole number to the decimal. 0 + 0.875 = 0.875. So, \(\frac{7}{8}\) = 0.875

Question 5. \(\frac{13}{20}\) _____

Answer: 0.65

Explanation: Use division to rename the fraction part as a decimal. \(\frac{13}{20}\) = 0.65 The quotient has 2 decimal places. Add the whole number to the decimal. 0 + 0.65 = 0.65. So, \(\frac{13}{20}\) = 0.65

Question 6. \(\frac{3}{25}\) _____

Answer: 0.12

Explanation: Use division to rename the fraction part as a decimal. \(\frac{3}{25}\) = 0.12 The quotient has 2 decimal places. Add the whole number to the decimal. 0 + 0.12 = 0.12. So, \(\frac{3}{25}\)= 0.12

On Your Own

Question 7. 0.27 \(\frac{□}{□}\)

Answer: \(\frac{27}{100}\)

Explanation: 0.27 is 0 ones and 27 hundredths. 27 hundredths = \(\frac{27}{100}\) Simplify using the GCF. The GCF of 27 and 100 is 1. Divide the numerator and the denominator by 1 \(\frac{27 ÷ 1}{100 ÷ 1}\) = \(\frac{27}{100}\)

Question 8. 0.055 \(\frac{□}{□}\)

Answer: \(\frac{11}{200}\)

Explanation: 0.055 is 0 ones and 55 thousandths. 55 thousandths = \(\frac{55}{1000}\) Simplify using the GCF. The GCF of 55 and 1000 is 5. Divide the numerator and the denominator by 5 \(\frac{55 ÷ 5}{1000 ÷ 5}\) = \(\frac{11}{200}\)

Question 9. 2.45 _____ \(\frac{□}{□}\)

Answer: \(\frac{9}{20}\)

Explanation: 2.45 is 2 ones and 45 hundredths. 45 hundredths = \(\frac{45}{100}\) Simplify using the GCF. The GCF of 45 and 100 is 5. Divide the numerator and the denominator by 1 \(\frac{45 ÷ 5}{100 ÷ 5}\) = \(\frac{9}{20}\)

Question 10. \(\frac{3}{8}\) _____

Answer: 0.375

Explanation: Use division to rename the fraction part as a decimal. \(\frac{3}{8}\) = 0.375 The quotient has 3 decimal places. Add the whole number to the decimal. 0 + 0.375 = 0.375. So, \(\frac{3}{8}\) = 0.375

Decimal Questions and Answers for Grade 6 Question 11. 3 \(\frac{1}{5}\) _____

Answer: 3.2

Explanation: Use division to rename the fraction part as a decimal. \(\frac{1}{5}\) = 0.2 The quotient has 1 decimal place. Add the whole number to the decimal. 3 + 0.2 = 3.2. So, 3 \(\frac{1}{5}\) = 3.2

Question 12. 2 \(\frac{11}{20}\) _____

Answer: 2.55

Explanation: Use division to rename the fraction part as a decimal. \(\frac{11}{20}\) = 0.55 The quotient has 2 decimal places. Add the whole number to the decimal. 2 + 0.55 = 2.55. So, 2 \(\frac{11}{20}\) = 2.55

Question 13. Point A Type below: __________

Answer: 0.2

Question 14. Point B Type below: __________

Answer: 0.9

Explanation: Point B is between 0.8 and 1.0. Every point is separated by 0.1. So, Point B is at 0.9

Question 15. Point C Type below: __________

Answer: 0.5

Explanation: Point C is between 0.4 and 0.6. Every point is separated by 0.1. So, Point C is at 0.5

Question 16. Point D Type below: __________

Answer: 0.1

Explanation: Point D is between 0 and 0.2. Every point is separated by 0.1. So, Point D is at 0.1

Problem Solving + Applications – Page No. 72

Question 17. Members of the Ozark Trail Hiking Club hiked a steep section of the trail in June and July. The table shows the distances club members hiked in miles. Write Maria’s July distance as a decimal. _____ miles

Answer: 2.625 miles

Explanation: Maria’s July distance = 2 \(\frac{5}{8}\) Use division to rename the fraction part as a decimal. \(\frac{5}{8}\) = 0.625 The quotient has 3 decimal places. Add the whole number to the decimal. 2 + 0.625 = 2.625. 2 \(\frac{5}{8}\) = 2.625

Question 18. How much farther did Zoey hike in June and July than Maria hiked in June and July? Explain how you found your answer. _____ miles

Answer: 0.7 miles

Explanation: Maria: June = 2.95, July = 2 \(\frac{5}{8}\) = 2.58 Zoey: June = 2.85, July = 3 \(\frac{3}{8}\) = 3.38 [2.85 + 3.38] – [2.95 + 2.58] = 0.7 miles

Question 19. What’s the Error? Tabitha’s hiking distance in July was 2 \(\frac{1}{5}\) miles. She wrote the distance as 2.02 miles. What error did she make? Type below: __________

Answer: Tabitha’s hiking distance in July was 2 \(\frac{1}{5}\) miles. 2 \(\frac{1}{5}\) Use division to rename the fraction part as a decimal. \(\frac{1}{5}\)  = 0.2 The quotient has 1 decimal place. Add the whole number to the decimal. 2 + 0.2 = 2.2. 2 \(\frac{1}{5}\) = 2.2 She wrote the distance as 2.02 miles in mistake.

Question 20. Use Patterns Write \(\frac{3}{8}, \frac{4}{8}, \text { and } \frac{5}{8}\) as decimals. What pattern do you see? Use the pattern to predict the decimal form of \(\frac{6}{8}\) and \(\frac{7}{8}\). Type below: __________

Answer: \(\frac{3}{8}, \frac{4}{8}, \text { and } \frac{5}{8}\) as decimals. 0.375, 0.5, 0.625 Each decimal is separated by 0.125. So, 6/8 = 0.625 + 0.125 = 0.75 7/8 = 0.75 + 0.125 = 0.875

Answer: Point A: 0.5 Point B: 0.7 Point C: 0.3 Point D: 0.8

Explanation: Every point is differentiated by 0.1 distance. The A is between 0.4 and 0.6 which is 0.5 The B is between 0.6 and 0.8 which is 0.7 The C is between 0.1 and 0.6 which is 0.53

Fractions and Decimals – Page No. 73

Question 1. 0.52 \(\frac{□}{□}\)

Answer: \(\frac{13}{25}\)

Explanation: 0.52 0.52 is 52 hundredths. 52 hundredths = \(\frac{52}{100}\) Simplify using the GCF. The GCF of 52 and 100 is 4. Divide the numerator and the denominator by 4 \(\frac{52 ÷ 4}{100 ÷ 4}\) = \(\frac{13}{25}\)

Question 2. 0.02 \(\frac{□}{□}\)

Answer: \(\frac{1}{50}\)

Explanation: 0.02 0.02 is 2 hundredths. 2 hundredths = \(\frac{2}{100}\) Simplify using the GCF. The GCF of 2 and 100 is 2. Divide the numerator and the denominator by 2 \(\frac{2 ÷ 2}{100 ÷ 2}\) = \(\frac{1}{50}\)

Question 3. 4.8 ______ \(\frac{□}{□}\)

Answer: \(\frac{4}{5}\)

Explanation: 4.8 4.8 is 4 ones and 8 tenths. 8 tenths = \(\frac{8}{10}\) Simplify using the GCF. The GCF of 8 and 10 is 2. Divide the numerator and the denominator by 2 \(\frac{8 ÷ 2}{10 ÷ 2}\) = \(\frac{4}{5}\)

Question 4. 6.025 ______ \(\frac{□}{□}\)

Answer: \(\frac{1}{40}\)

Explanation: 6.025 is 6 ones and 25 thousandths. 25 thousandths = \(\frac{25}{1000}\) Simplify using the GCF. The GCF of 25 and 1000 is 25. Divide the numerator and the denominator by 25 \(\frac{25 ÷ 25}{1000 ÷ 25}\) = \(\frac{1}{40}\)

Question 5. \(\frac{17}{25}\) ______

Answer: 0.68

Explanation: Use division to rename the fraction part as a decimal. 17/25 = 0.68 The quotient has 2 decimal places. Add the whole number to the decimal. 0 + 0.68 = 0.68. So, \(\frac{17}{25}\) = 0.68

Question 6. \(\frac{11}{20}\) ______

Answer: 0.55

Explanation: Use division to rename the fraction part as a decimal. 11/20 = 0.55 The quotient has 2 decimal places. Add the whole number to the decimal. 0 + 0.55 = 0.55. So, \(\frac{11}{20}\) = 0.55

Question 7. 4 \(\frac{13}{20}\) ______

Answer: 4.65

Explanation: Use division to rename the fraction part as a decimal. \(\frac{13}{20}\) = 0.65 The quotient has 2 decimal places. Add the whole number to the decimal. 4 + 0.65 = 4.65. So, 4 \(\frac{13}{20}\) = 4.65

Question 8. 7 \(\frac{3}{8}\) ______

Answer: 7.375

Explanation: Use division to rename the fraction part as a decimal. \(\frac{3}{8}\) = 0.375 The quotient has 3 decimal places. Add the whole number to the decimal. 7 + 0.375 = 7.375. So, 7 \(\frac{3}{8}\) = 7.375

Question 9. Point A Type below: __________

Answer: 0.4

Explanation:

Point A is between 0 and 0.5. Every point is separated by 0.1. So, Point A is at 0.4

Question 10. Point D Type below: __________

Answer: 1.9

Explanation: Point D is between 1.5 and 2. Every point is separated by 0.1. So, Point D is at 1.9

Question 11. Point C Type below: __________

Answer: 1.2

Explanation: Point C is between 1 and 1.5. Every point is separated by 0.1. So, Point C is at 1.2

Question 12. Point B Type below: __________

Answer: 0.6

Explanation: Point C is between 0.5 and 1. Every point is separated by 0.1. So, Point C is at 0.6

Problem Solving

Question 13. Grace sold \(\frac{5}{8}\) of her stamp collection. What is this amount as a decimal? ______

Answer: 0.625

Explanation: Grace sold \(\frac{5}{8}\) of her stamp collection. Use division to rename the fraction part as a decimal. \(\frac{5}{8}\)  = 0.625 The quotient has 3 decimal places. Add the whole number to the decimal. 0 + 0.625 = 0.625. So, \(\frac{5}{8}\) = 0.625

Question 14. What if you scored an 0.80 on a test? What fraction of the test, in simplest form, did you answer correctly? \(\frac{□}{□}\)

Explanation: 0.80 is 0 ones and 8 tenths. 8 tenths = \(\frac{8}{10}\) Simplify using the GCF. The GCF of 8 and 10 is 2. Divide the numerator and the denominator by 2 \(\frac{8 ÷ 2}{10 ÷ 2}\) = \(\frac{4}{5}\)

Chapter 2 Fractions Decimals and Percents Question 15. What fraction in simplest form is equivalent to 0.45? What decimal is equivalent to \(\frac{17}{20}\)? Explain how you found your answers. Type below: __________

Answer: 0.45 is 0 ones and 45 hundredths. 45 hundredths = \(\frac{45}{100}\) Simplify using the GCF. The GCF of 45 and 100 is 5. Divide the numerator and the denominator by 5 \(\frac{45 ÷ 5}{100 ÷ 5}\) = \(\frac{9}{20}\) \(\frac{17}{20}\) Use division to rename the fraction part as a decimal. \(\frac{17}{20}\) = 0.85 The quotient has 2 decimal places. Add the whole number to the decimal. 0 + 0.85 = 0.85. So, \(\frac{17}{20}\) = 0.85

Lesson Check – Page No. 74

Question 1. After a storm, Michael measured 6 \(\frac{7}{8}\) inches of snow. What is this amount as a decimal? ______ inches

Answer: 6.875 inches

Explanation: Michael measured 6 \(\frac{7}{8}\) inches of snow. Use division to rename the fraction part as a decimal. \(\frac{7}{8}\) = 0.875 The quotient has 3 decimal places. Add the whole number to the decimal. 6 + 0.875 = 6.875. So, 6 \(\frac{7}{8}\) = 6.875.

Question 2. A recipe calls for 3.75 cups of flour. What is this amount as a mixed number in simplest form? ______ \(\frac{□}{□}\) cups

Answer: 3 \(\frac{3}{4}\) cups

Explanation: A recipe calls for 3.75 cups of flour. 3 + 0.75 0.75 is 0 ones and 75 hundredths. 75 hundredths = \(\frac{75}{100}\) Simplify using the GCF. The GCF of 75 and 100 is 25. Divide the numerator and the denominator by 25 \(\frac{75 ÷ 25}{100 ÷ 25}\) = \(\frac{3}{4}\) 3 \(\frac{3}{4}\)

Spiral Review

Question 3. Gina bought 2.3 pounds of red apples and 2.42 pounds of green apples. They were on sale for $0.75 a pound. How much did the apples cost altogether? $ ______

Answer: $3.54

Explanation: Gina bought 2.3 pounds of red apples and 2.42 pounds of green apples. They were on sale for $0.75 a pound. $0.75 x 2.3 = 1.725 $0.75 x 2.42 = 1.815 1.725 + 1.815 = 3.54 So the apples cost $3.54

Question 4. Ken has 4.66 pounds of walnuts, 2.1 pounds of cashews, and 8 pounds of peanuts. He mixes them together and divides them equally among 18 bags. How many pounds of nuts are in each bag? ______ pounds

Answer: 0.82 pounds

Explanation: Ken has 4.66 pounds of walnuts, 2.1 pounds of cashews, and 8 pounds of peanuts. 4.66 + 2.1 + 8 = 14.76 He mixes them together and divides them equally among 18 bags. 14.76/18 = 0.82

Question 5. Mia needs to separate 270 blue pens and 180 red pens into packs. Each pack will have the same number of blue pens and the same number of red pens. What is the greatest number of packs she can make? How many red pens and how many blue pens will be in each pack? Type below: __________

Answer: There are 2 red pens and 3 blue pens in each pack.

Explanation: Mia needs to separate 270 blue pens and 180 red pens into packs. The GCF of 270 and 180 is 90 The greatest number of packs she can make is 90. Divide the total number of red pens by the total number of packs. 180/90 = 2 Divide the total number of blue pens by the total number of packs. 270/90 = 3 There are 2 red pens and 3 blue pens in each pack.

Question 6. Evan buys 19 tubes of watercolor paint for $50.35. What is the cost of each tube of paint? $ ______

Answer: $2.65

Explanation: Evan buys 19 tubes of watercolor paint for $50.35. $50.35/19 = $2.65

Share and Show – Page No. 77

Order from least to greatest.

Question 1. \(3 \frac{3}{6}, 3 \frac{5}{8}, 2 \frac{9}{10}\) Type below: __________

Answer: 2 \(\frac{9}{10}\) < 3 \(\frac{3}{6}\) < 3 \(\frac{5}{8}\)

Explanation: \(3 \frac{3}{6}, 3 \frac{5}{8}, 2 \frac{9}{10}\) Compare the whole numbers first. 2 < 3 If the whole numbers are the same, compare the fractions. 3 \(\frac{3}{6}\), 3 \(\frac{5}{8}\) 6 and 8 are multiples of 48. So, 48 is a common denominator. 3 \(\frac{3 x 8}{6 x 8}\) = 3 \(\frac{24}{48}\), 3 \(\frac{5 x 6}{8 x 6}\) = 3 \(\frac{30}{48}\) 3 \(\frac{24}{48}\) < 3 \(\frac{30}{48}\) So, 3 \(\frac{3}{6}\) < 3 \(\frac{5}{8}\) Order the fractions from least to greatest. 2 \(\frac{9}{10}\) < 3 \(\frac{3}{6}\) < 3 \(\frac{5}{8}\)

Write <, >, or =.

Question 2. 0.8 _____ \(\frac{4}{12}\)

Answer: 0.8 < latex]\frac{4}{12}[/latex]

Explanation: Write the decimal form of \(\frac{4}{12}\) = 0.3333 0.8 > 0.333 So, 0.8 < latex]\frac{4}{12}[/latex]

Question 3. 0.22 _____ \(\frac{1}{4}\)

Answer: 0.22 < \(\frac{1}{4}\)

Explanation: Write the decimal form of \(\frac{1}{4}\) = 0.25 0.22 < 0.25 So, 0.22 < \(\frac{1}{4}\)

Question 4. \(\frac{1}{20}\) _____ 0.06

Answer: \(\frac{1}{20}\) < 0.06

Explanation: Write the decimal form of \(\frac{1}{20}\) = 0.05 0.05 < 0.06 So, \(\frac{1}{20}\) < 0.06

Use a number line to order from least to greatest.

Question 5. \(1 \frac{4}{5}, 1.25, 1 \frac{1}{10}\) Type below: __________

Answer: 1\(\frac{1}{10}\), 1.25, 1\(\frac{4}{5}\)

Explanation: Write the decimal form of 1\(\frac{4}{5}\) = 1.8 Write the decimal form of 1\(\frac{1}{10}\) = 1.1 1.8, 1.25, 1.1 Locate each decimal on a number line. So, from least to greatest, the order is 1.1, 1.25, 1.8 1\(\frac{1}{10}\), 1.25, 1\(\frac{4}{5}\)

Question 6. 0.6, \(\frac{4}{5}\), 0.75 Type below: __________

Answer: 0.6, 0.75, \(\frac{4}{5}\)

Explanation: Write the decimal form of \(\frac{4}{5}\) = 0.8 0.6, 0.8, 0.75 Compare decimals. All ones are equal. Compare tenths: 6 < 7 < 8 So, from least to greatest, the order is 0.6, 0.75, 0.8 So, 0.6, 0.75, \(\frac{4}{5}\)

Practice and Homework Lesson 2.2 Answer Key Question 7. \(\frac{1}{2}\), \(\frac{2}{5}\), \(\frac{7}{15}\) Type below: __________

Answer: \(\frac{2}{5}\), \(\frac{7}{15}\), \(\frac{1}{2}\)

Explanation: Write the decimal form of \(\frac{1}{2}\) = 0.5 Write the decimal form of \(\frac{2}{5}\) = 0.4 Write the decimal form of \(\frac{7}{15}\) = 0.466 0.5, 0.4, 0.466 Compare decimals. All ones are equal. Compare tenths: 4 < 5 Compare hundredths of 0.4 and 0.466; 0 < 6 So, from least to greatest, the order is 0.4 < 0.466 < 0.5 So, \(\frac{2}{5}\), \(\frac{7}{15}\), \(\frac{1}{2}\)

Question 8. 5 \(\frac{1}{2}\), 5.05, 5 \(\frac{5}{9}\) Type below: __________

Answer: 5.05, 5 \(\frac{1}{2}\), 5 \(\frac{5}{9}\)

Explanation: Write the decimal form of 5 \(\frac{1}{2}\) = 5.5 Write the decimal form of 5 \(\frac{5}{9}\) = 5.555 5.5, 5.05, 5.5555 Compare decimals. All ones are equal. Compare tenths: 0 < 5 Compare hundredths of 5.5 and 5.55; 0 < 5 So, from least to greatest, the order is 5.05 < 5.5 < 5.55 So, 5.05, 5 \(\frac{1}{2}\), 5 \(\frac{5}{9}\)

Question 9. \(\frac{5}{7}\), \(\frac{5}{6}\), \(\frac{5}{12}\) Type below: __________

Answer: \(\frac{5}{12}\), \(\frac{5}{7}\), \(\frac{5}{6}\)

Explanation: \(\frac{5}{7}\), \(\frac{5}{6}\), \(\frac{5}{12}\) To compare fractions with the same numerators, compare the denominators. So, from least to greatest, the order is \(\frac{5}{12}\), \(\frac{5}{7}\), \(\frac{5}{6}\)

Question 10. \(\frac{7}{15}\) _____ \(\frac{7}{10}\)

Answer: \(\frac{7}{15}\) < \(\frac{7}{10}\)

Explanation: \(\frac{7}{15}\) and \(\frac{7}{10}\) To compare fractions with the same numerators, compare the denominators. So, \(\frac{7}{15}\) < \(\frac{7}{10}\)

Question 11. \(\frac{1}{8}\) _____ 0.125

Answer: \(\frac{1}{8}\) = 0.125

Explanation: Write the decimal form of \(\frac{1}{8}\) = 0.125 0.125 = 0.125

Question 12. 7 \(\frac{1}{3}\) _____ 6 \(\frac{2}{3}\)

Answer: 7 \(\frac{1}{3}\) > 6 \(\frac{2}{3}\)

Explanation: Compare the whole numbers first. 7 > 6. So, 7 \(\frac{1}{3}\) > 6 \(\frac{2}{3}\)

Question 13. 1 \(\frac{2}{5}\) _____ 1 \(\frac{7}{15}\)

Answer: 1 \(\frac{2}{5}\) < 1 \(\frac{7}{15}\)

Explanation: 1 \(\frac{2}{5}\) _____ 1 \(\frac{7}{15}\) If the whole numbers are the same, compare the fractions. Compare \(\frac{2}{5}\) and \(\frac{7}{15}\) 5 and 15 are multiples of 15. So, \(\frac{2 x 3}{5 x 3}\) = \(\frac{6}{15}\) \(\frac{6}{15}\) < \(\frac{7}{15}\) Use common denominators to write equivalent fractions. 1 \(\frac{2}{5}\) < 1 \(\frac{7}{15}\)

Question 14. Darrell spent 3 \(\frac{2}{5}\) hours on a project for school. Jan spent 3 \(\frac{1}{4}\) hours and Maeve spent 3.7 hours on the project. Who spent the least amount of time? Show how you found your answer. Then describe another possible method. Type below: __________

Answer: Jan spent the least amount of time.

Explanation: Darrell spent 3 \(\frac{2}{5}\) hours on a project for school. Jan spent 3 \(\frac{1}{4}\) hours and Maeve spent 3.7 hours on the project. Write the decimal form of 3 \(\frac{2}{5}\) = 3.4 Write the decimal form of 3 \(\frac{1}{4}\) = 3.25 3.4, 3.25, 3.7 3.25 is the least one. So, Jan spent the least amount of time.

Problem Solving + Applications – Page No. 78

Question 15. In one week, Altoona, PA, and Bethlehem, PA, received snowfall every day, Monday through Friday. On which days did Altoona receive over 0.1 inch more snow than Bethlehem? Type below: __________

Answer: Altoona received over 1 inch more snow than Bethlehem on Friday

Explanation: Altoona (converted to decimal form): 2.25, 3.25, 2.625, 4.6, 4.75 Bethlehem: 2.6, 3.2, 2.5, 4.8, 2.7 Altoona received over 1 inch more snow than Bethlehem on Friday

Question 16. What if Altoona received an additional 0.3 inch of snow on Thursday? How would the total amount of snow in Altoona compare to the amount received in Bethlehem that day? Type below: __________

Answer: Altoona received 0.1 inches more snow than Bethlehem on Thursday

Explanation: Altoona received an additional 0.3 inch of snow on Thursday = 4.6 + 0.3 = 4.9 Bethlehem received on Thursday = 4.8 Altoona received 0.1 inches more snow than Bethlehem on Thursday

Question 17. Explain two ways you could compare the snowfall amounts in Altoona and Bethlehem on Monday. Type below: __________

Explanation: Altoona received on Monday = 2.25 Bethlehem received on Monday = 2.6 Bethlehem received 0.35 inches more snow than Altoona on Monday. As the whole numbers are equal compare 1/4 and 0.6. 0.25 < 0.6 So, Altoona received less snow compared to Bethlehem on Monday.

Question 18. Explain how you could compare the snowfall amounts in Altoona on Thursday and Friday. Type below: __________

Answer: Altoona received on Thursday = 4.6 Altoona received on Friday = 4.75 4.6 < 4.75 Altoona received less snow on Thursday compared to Friday.

Answer: 1/3, 0.39, 2/5, 0.45

Explanation: 1/3 = 0.333 0.45 0.39 2/5 = 0.4 Compare tenths: 3 < 4 Compare hundredths: 0.33 < 0.39 0.4 < 0.45 So, 1/3, 0.39, 2/5, 0.45

Compare and Order Fractions and Decimals – Page No. 79

Write <, >, =.

Question 1. 0.64 _____ \(\frac{7}{10}\)

Answer: 0.64 < \(\frac{7}{10}\)

Explanation: Write the decimal form of \(\frac{7}{10}\) = 0.7 Compare tenths: 6 < 7 So, 0.64 < 0.7 0.64 < \(\frac{7}{10}\)

Question 2. 0.48 _____ \(\frac{6}{15}\)

Answer: 0.48 > \(\frac{6}{15}\)

Explanation: Write the decimal form of \(\frac{6}{15}\) = 0.4 Compare hundredths: 0.48 > 0.4 0.48 > \(\frac{6}{15}\)

Question 3. 0.75 _____ \(\frac{7}{8}\)

Answer: 0.75 < \(\frac{7}{8}\)

Explanation: Write the decimal form of \(\frac{7}{8}\) = 0.875 Compare tenths: 7 < 8 0.75 < \(\frac{7}{8}\)

Practice and Homework Lesson 2.2 Question 4. 7 \(\frac{1}{8}\) _____ 7.025

Answer: 7 \(\frac{1}{8}\) > 7.025

Explanation: Write the decimal form of 7 \(\frac{1}{8}\) = 7.125 Compare tenths: 1 > 0 7 \(\frac{1}{8}\) > 7.025

Question 5. \(\frac{7}{15}\), 0.75, \(\frac{5}{6}\) Type below: __________

Answer: \(\frac{7}{15}\), 0.75, \(\frac{5}{6}\)

Explanation: Write the decimal form of \(\frac{7}{15}\) = 0.466 0.75 Write the decimal form of \(\frac{5}{6}\) = 0.833 Order from least to greatest: \(\frac{7}{15}\), 0.75, \(\frac{5}{6}\)

Question 6. 0.5, 0.41, \(\frac{3}{5}\) Type below: __________

Answer: 0.41, 0.5, \(\frac{3}{5}\)

Explanation: Write the decimal form of \(\frac{3}{5}\) = 0.6 Compare tenths: 0.41, 0.5, 0.6 Order from least to greatest: 0.41, 0.5, \(\frac{3}{5}\)

Question 7. 3.25, 3 \(\frac{2}{5}\), 3 \(\frac{3}{8}\) Type below: __________

Answer: 3.25, 3 \(\frac{2}{5}\), 3 \(\frac{3}{8}\)

Explanation: Write the decimal form of 3 \(\frac{2}{5}\) = 3.4 Write the decimal form of 3 \(\frac{3}{8}\) = 3.375 Compare tenths: Order from least to greatest: 3.25, 3 \(\frac{2}{5}\), 3 \(\frac{3}{8}\)

Question 8. 0.9, \(\frac{8}{9}\), 0.86 Type below: __________

Answer: 0.86, \(\frac{8}{9}\), 0.9

Explanation: Write the decimal form of \(\frac{8}{9}\) = 0.88 Compare tenths: 0.86, 0.88, 0.9 Order from least to greatest: 0.86, \(\frac{8}{9}\), 0.9

Order from greatest to least.

Question 9. 0.7, \(\frac{7}{9}\), \(\frac{7}{8}\) Type below: __________

Answer: \(\frac{7}{8}\), \(\frac{7}{9}\), 0.7

Explanation: 0.7 = 7/10 To compare fractions with the same numerators, compare the denominators. 7/10, 7/9, 7/8 Order from greatest to least: 7/8, 7/9, 7/10

Question 10. 0.2, 0.19, \(\frac{3}{5}\) Type below: __________

Answer: \(\frac{3}{5}\), 0.2, 0.19

Explanation: Write the decimal form of \(\frac{3}{5}\) = 0.6 Compare tenths: 0.6, 0.2, 0.19 Order from greatest to least: \(\frac{3}{5}\), 0.2, 0.19

Question 11. 6\(\frac{1}{20}\), 6.1, 6.07 Type below: __________

Explanation: Write the decimal form of 6\(\frac{1}{20}\) = 121/20 = 6.05 Compare tenths: 6.1, 6.07, 6.05 Order from greatest to least: 6.1, 6.07, 6\(\frac{1}{20}\)

Question 12. 2 \(\frac{1}{2}\), 2.4, 2.35, 2 \(\frac{1}{8}\) Type below: __________

Answer: 2 \(\frac{1}{2}\), 2.4, 2.35, 2 \(\frac{1}{8}\)

Explanation: Write the decimal form of 2 \(\frac{1}{2}\) = 2.5 Write the decimal form of 2 \(\frac{1}{8}\) = 2.125 Compare tenths: 2.5, 2.4, 2.35, 2.125 Order from greatest to least: 2 \(\frac{1}{2}\), 2.4, 2.35, 2 \(\frac{1}{8}\)

Question 13. One day it snowed 3 \(\frac{3}{8}\) inches in Altoona and 3.45 inches in Bethlehem. Which city received less snow that day? __________

Answer: Altoona

Explanation: One day it snowed 3 \(\frac{3}{8}\) inches in Altoona and 3.45 inches in Bethlehem. Write the decimal form of 3 \(\frac{3}{8}\) = 27/8 = 3.375 3.375 < 3.45. Altoona received less snow that day

Question 14. Malia and John each bought 2 pounds of sunflower seeds. Each ate some seeds. Malia has 1 \(\frac{1}{3}\) pounds left, and John has 1 \(\frac{2}{5}\) pounds left. Who ate more sunflower seeds? __________

Answer: Malia

Explanation: Malia and John each bought 2 pounds of sunflower seeds. Each ate some seeds. Malia has 1 \(\frac{1}{3}\) pounds left, and John has 1 \(\frac{2}{5}\) pounds left. 2 – 1 \(\frac{1}{3}\) = 0.667 2 – 1 \(\frac{2}{5}\) = 0.6 0.667 > 0.6 So, Malia ate more sunflower seeds

Question 15. Explain how you would compare the numbers 0.4 and \(\frac{3}{8}\). Type below: __________

Answer: Write the decimal form of \(\frac{3}{8}\) = 0.375 Compare tenths: 0.4 > 0.375

Lesson Check – Page No. 80

Question 1. Andrea has 3 \(\frac{7}{8}\) yards of purple ribbon, 3.7 yards of pink ribbon, and 3 \(\frac{4}{5}\) yards of blue ribbon. List the numbers in order from least to greatest. Type below: __________

Answer: Andrea has 3 \(\frac{7}{8}\) yards of purple ribbon, 3.7 yards of pink ribbon, and 3 \(\frac{4}{5}\) yards of blue ribbon. Write the decimal form of 3 \(\frac{7}{8}\) = 3.875 3.7 Write the decimal form of 3 \(\frac{4}{5}\) = 3.8 Least to greatest : 3.7, 3 \(\frac{4}{5}\), 3 \(\frac{7}{8}\)

Question 2. Nassim completed \(\frac{18}{25}\) of the math homework. Kara completed 0.7 of it. Debbie completed \(\frac{5}{8}\) of it. List the numbers in order from greatest to least. Type below: __________

Answer: $1.39, $0.70, $0.63

Explanation: Nassim completed \(\frac{18}{25}\) of the math homework. Kara completed 0.7 of it. Debbie completed \(\frac{5}{8}\) of it. Write the decimal form of 18/25 = 1.39 0.7 Write the decimal form of 5/8 = 0.63 They are now in order from greatest to least. Think of the amounts as money: $1.39, $0.70, $0.63

Question 3. Tyler bought 3 \(\frac{2}{5}\) pounds of oranges. Graph 3 \(\frac{2}{5}\) on a number line and write this amount using a decimal. Type below: __________

Question 4. At the factory, a baseball card is placed in every 9th package of cereal. A football card is placed in every 25th package of the cereal. What is the first package that gets both a baseball card and a football card? Type below: __________

Answer: 225th package

Explanation: Look for the first number where both 25 and 9 are a factor of. 25 x 1 = 25 which isn’t a factor of 9, so it won’t be 25. 25 x 2 = 50, which isn’t a factor of 9. 75 is not a factor of 9. (you know because you don’t get a whole number when you divide 75 into 9.) 100 is not a factor of 9, nor is 125, 150, 175, or 200. However, 225 is a factor of both 25 and 9. This makes sense because 25 x 9 is 225. This means that the first package with both will be the 225th package.

Question 5. $15.30 is divided among 15 students. How much does each student receive? $ _____

Answer: $1.02

Explanation: $15.30 is divided among 15 students. $15.30/15 = $1.02 Each student receives $1.02

Question 6. Carrie buys 4.16 pounds of apples for $5.20. How much does 1 pound cost? $ _____

Answer: $1.25

Explanation: Carrie buys 4.16 pounds of apples for $5.20. $5.20/4.16 = $1.25 1 pound cost = $1.25

Share and Show – Page No. 83

Find the product. Write it in simplest form.

Question 1. 6 × \(\frac{3}{8}\) \(\frac{□}{□}\)

Answer: \(\frac{9}{4}\)

Explanation: \(\frac{6 × 3}{1 × 8}\) \(\frac{18}{8}\) Simplify using the GCF. The GCF of 18 and 8 is 2. Divide the numerator and the denominator by 2. \(\frac{18 ÷ 2}{8 ÷ 2}\) = \(\frac{9}{4}\)

Question 2. \(\frac{3}{8}\) × \(\frac{8}{9}\) \(\frac{□}{□}\)

Answer: \(\frac{1}{3}\)

Explanation: Multiply the numerators and Multiply the denominators. \(\frac{3 × 8}{8 × 9}\) = \(\frac{24}{72}\) Simplify using the GCF. The GCF of 24 and 72 is 24. Divide the numerator and the denominator by 24. \(\frac{24 ÷ 24}{72 ÷ 24}\) = \(\frac{1}{3}\)

Practice and Homework Lesson 2.3 Answer Key Question 3. Sam and his friends ate 3 \(\frac{3}{4}\) bags of fruit snacks. If each bag contained 2 \(\frac{1}{2}\) ounces, how many ounces of fruit snacks did Sam and his friends eat? \(\frac{□}{□}\)

Answer: \(\frac{75}{8}\) ounces

Explanation: Sam and his friends ate 3 \(\frac{3}{4}\) bags of fruit snacks. If each bag contained 2 \(\frac{1}{2}\) ounces 3 \(\frac{3}{4}\) x 2 \(\frac{1}{2}\) \(\frac{15}{4}\) x \(\frac{5}{2}\) \(\frac{15 x 5}{4 x 2}\) = \(\frac{75}{8}\)

Attend to Precision Algebra Evaluate using the order of operations.

Write the answer in the simplest form.

Question 4. \(\left(\frac{3}{4}-\frac{1}{2}\right) \times \frac{3}{5}\) \(\frac{□}{□}\)

Answer: \(\frac{3}{20}\)

Explanation: \(\left(\frac{3}{4}-\frac{1}{2}\right) \times \frac{3}{5}\) Perform operations in parentheses. \(\frac{3}{4}\) – \(\frac{1}{2}\) = \(\frac{1}{4}\) \(\frac{1}{4}\) x \(\frac{3}{5}\) = \(\frac{1 x 3}{4 x 5}\) = \(\frac{3}{20}\)

Question 5. \(\frac{1}{3}+\frac{4}{9} \times 12\) \(\frac{□}{□}\)

Answer: \(\frac{28}{3}\)

Explanation: \(\frac{1}{3}\) + \(\frac{4}{9}\) = \(\frac{7}{9}\) \(\frac{7 x 12}{9 x 1}\) = \(\frac{84}{9}\) Simplify using the GCF. The GCF of 84 and 9 is 3. Divide the numerator and the denominator by 3. \(\frac{84 ÷ 3}{9 ÷ 3}\) = \(\frac{28}{3}\)

Question 6. \(\frac{5}{8} \times \frac{7}{10}-\frac{1}{4}\) \(\frac{□}{□}\)

Answer: \(\frac{11}{16}\)

Explanation: \(\frac{5 x 7}{8 x 10}\) = \(\frac{35}{80}\) \(\frac{35}{80}\) – \(\frac{1}{4}\) = \(\frac{11}{16}\)

Question 7. 3 × (\(\frac{5}{18}\) + \(\frac{1}{6}\)) + \(\frac{2}{5}\) \(\frac{□}{□}\)

Answer: \(\frac{38}{15}\)

Explanation: 3 x \(\frac{4}{9}\) + \(\frac{2}{5}\) 3 x \(\frac{38}{45}\) = \(\frac{38}{15}\)

Practice: Copy and Solve Find the product. Write it in simplest form.

Question 8. \(1 \frac{2}{3} \times 2 \frac{5}{8}\) \(\frac{□}{□}\)

Answer: \(\frac{35}{8}\)

Explanation: 1 \(\frac{2}{3}\) = \(\frac{5}{3}\) 2 \(\frac{5}{8}\) = \(\frac{21}{8}\) \(\frac{5 × 21}{3 × 8}\) = \(\frac{105}{24}\) Simplify using the GCF The GCF of 105 and 24 is 3. Divide the numerator and the denominator by 3. \(\frac{105 ÷ 3}{24 ÷ 3}\) = \(\frac{35}{8}\)

Question 9. \(\frac{4}{9} \times \frac{4}{5}\) \(\frac{□}{□}\)

Answer: \(\frac{16}{45}\)

Explanation: \(\frac{4 × 4}{9 × 5}\) = \(\frac{16}{45}\)

Question 10. \(\frac{1}{6} \times \frac{2}{3}\) \(\frac{□}{□}\)

Answer: \(\frac{1}{9}\)

Explanation: \(\frac{1 × 2}{6 × 3}\) = \(\frac{2}{18}\) Simplify using the GCF The GCF of 2 and 18 is 2. Divide the numerator and the denominator by 2. \(\frac{2 ÷ 2}{18 ÷ 2}\) = \(\frac{1}{9}\)

Question 11. \(4 \frac{1}{7} \times 3 \frac{1}{9}\) \(\frac{□}{□}\)

Answer: \(\frac{116}{7}\)

Explanation: 4\(\frac{1}{7}\) = \(\frac{29}{7}\) 3\(\frac{1}{9}\) = \(\frac{28}{9}\) \(\frac{29 × 28}{7 × 9}\) = \(\frac{812}{63}\) Simplify using the GCF The GCF of 812 and 63 is 7. Divide the numerator and the denominator by 7. \(\frac{812 ÷ 7}{63 ÷ 7}\) = \(\frac{116}{7}\)

Question 12. \(\frac{5}{6}\) of the 90 pets in the pet show are cats. \(\frac{4}{5}\) of the cats are calico cats. What fraction of the pets are calico cats? How many of the pets are calico cats? Type below: __________

Answer: 60 calico cats

Explanation: 5/6 x 90 = 450/6 = 150/2 150/2 x 4/5 = 60

Question 13. Five cats each ate \(\frac{1}{4}\) cup of cat food. Four other cats each ate \(\frac{1}{3}\) cup of cat food. How much food did the nine cats eat? Type below: __________

Answer: \(\frac{31}{12}\)

Explanation: 5 x 1/4 = 5/4 4 x 1/3 = 4/3 5/4 + 4/3 = 31/12

Question 14. \(\frac{1}{4} \times\left(\frac{3}{9}+5\right)\) \(\frac{□}{□}\)

Answer: \(\frac{4}{3}\)

Explanation: 3/9 + 5 = 16/3 1/4 x 16/3 1 x 16 = 16 4 x 3 = 12 16/12 Simplify using the GCF The GCF of 16 and 12 is 4. Divide the numerator and the denominator by 4. \(\frac{16 ÷ 4}{12÷ 4}\) = \(\frac{4}{3}\)

Question 15. \(\frac{9}{10}-\frac{3}{5} \times \frac{1}{2}\) \(\frac{□}{□}\)

Explanation: 3/5 x 1/2 = 3/10 9/10 – 3/10 = 6/10 Simplify using the GCF The GCF of 6 and 10 is 2. Divide the numerator and the denominator by 2. \(\frac{6 ÷ 2}{10 ÷ 2}\) = \(\frac{3}{5}\)

Question 16. \(\frac{4}{5}+\left(\frac{1}{2}-\frac{3}{7}\right) \times 2\) \(\frac{□}{□}\)

Answer: \(\frac{33}{35}\)

Explanation: 1/2 – 3/7 = 1/14 1/14 x 2 = 1/7 4/5 + 1/7 = 33/35

Question 17. \(15 \times \frac{3}{10}+\frac{7}{8}\) \(\frac{□}{□}\)

Answer: \(\frac{141}{8}\)

Explanation: 3/10 + 7/8 = 47/40 15 x 47/40 = 141/8 \(\frac{141}{8}\)

Page No. 84

Question 18. Write and solve a word problem for the expression \(\frac{1}{4} \times \frac{2}{3}\). Show your work. Type below: __________

Answer: \(\frac{1}{6}\)

Explanation: \(\frac{1}{4} \times \frac{2}{3}\) = \(\frac{1 X 2}{4 X 3}\) = \(\frac{2}{12}\) Simplify using the GCF The GCF of 2 and 12 is 2. Divide the numerator and the denominator by 2. \(\frac{2 ÷ 2}{12 ÷ 2}\) = \(\frac{1}{6}\)

Question 19. Michelle has a recipe that asks for 2 \(\frac{1}{2}\) cups of vegetable oil. She wants to use \(\frac{2}{3}\) that amount of oil and use applesauce to replace the rest. How much applesauce will she use? Type below: __________

Answer: \(\frac{10}{6}\)

Explanation: 2 1/2 * 2/3 = 5/2 * 2/3 = 10/6 She will use 10/6 or 1 2/3 cups of vegetable oil

Question 20. Cara’s muffin recipe asks for 1 \(\frac{1}{2}\) cups of flour for the muffins and \(\frac{1}{4}\) cup of flour for the topping. If she makes \(\frac{1}{2}\) of the original recipe, how much flour will she use for the muffins and topping? Type below: __________

Answer: Cara will use 1\(\frac{1}{8}\) cups of flour.

Explanation: For first we will find how many cups of flours need to makes the original recipe. Cara uses 1 1/2 cups of flour for the muffins and 1/4 cup off flour for the topping. So, 1 1/2 + 1/4 cups of flour to make the original recipe. 1 1/2 = 3/2 3/2 + 1/4 = 7/4 To make the original recipe Cara needs 7/4 cups of flour. If she makes \(\frac{1}{2}\) of the original recipe, then 7/4 x 1/2 = 7/8 = 1 1/8 Cara will use 1 1/8 cups of flour.

Multiply Fractions – Page No. 85

Question 1. \(\frac{4}{5} \times \frac{7}{8}\) \(\frac{□}{□}\)

Answer: \(\frac{7}{10}\)

Explanation: Multiply the numerators and Multiply the denominators. \(\frac{4 × 7}{5 × 8}\) = \(\frac{28}{40}\) Simplify using the GCF. The GCF of 28 and 40 is 4. Divide the numerator and the denominator by 4. \(\frac{28 ÷ 4}{40 ÷ 4}\) = \(\frac{7}{10}\)

Question 2. \(\frac{1}{8} \times 20\) \(\frac{□}{□}\)

Answer: \(\frac{5}{2}\)

Explanation: \(\frac{1 × 20}{1 × 8}\) \(\frac{20}{8}\) Simplify using the GCF. The GCF of 20 and 8 is 4. Divide the numerator and the denominator by 4. \(\frac{20 ÷ 4}{8 ÷ 4}\) = \(\frac{5}{2}\)

Question 3. \(\frac{4}{5} \times \frac{3}{8}\) \(\frac{□}{□}\)

Answer: \(\frac{3}{10}\)

Explanation: Multiply the numerators and Multiply the denominators. \(\frac{4 × 3}{5 × 8}\) = \(\frac{12}{40}\) Simplify using the GCF. The GCF of 12 and 40 is 4. Divide the numerator and the denominator by 4. \(\frac{12 ÷ 4}{40 ÷ 4}\) = \(\frac{3}{10}\)

Question 4. \(1 \frac{1}{8} \times \frac{1}{9}\) \(\frac{□}{□}\)

Answer: \(\frac{1}{8}\)

Explanation: 1\(\frac{1}{8}\) = \(\frac{9}{8}\) Multiply the numerators and Multiply the denominators. \(\frac{9 × 1}{8 × 9}\) = \(\frac{9}{72}\) Simplify using the GCF. The GCF of 9 and 72 is 9. Divide the numerator and the denominator by 9. \(\frac{9 ÷ 9}{72 ÷ 9}\) = \(\frac{1}{8}\)

Question 5. \(\frac{3}{4} \times \frac{1}{3} \times \frac{2}{5}\) \(\frac{□}{□}\)

Answer: \(\frac{1}{10}\)

Explanation: Multiply the numerators and Multiply the denominators. \(\frac{3 × 1 × 2}{4 × 3 × 5}\) = \(\frac{6}{60}\) Simplify using the GCF. The GCF of 6 and 60 is 6. Divide the numerator and the denominator by 6. \(\frac{6 ÷ 6}{60 ÷ 6}\) = \(\frac{1}{10}\)

Question 6. Karen raked \(\frac{3}{5}\) of the yard. Minni raked \(\frac{1}{3}\) of the amount Karen raked. How much of the yard did Minni rake? \(\frac{□}{□}\)

Explanation: Minni raked 1/5 of the yard. So, minni raked 3/5 of 1/3 means 3/5 x 1/3 Multiply the numerators and Multiply the denominators. \(\frac{3 × 1}{5 × 3}\) = \(\frac{3}{15}\) Simplify using the GCF. The GCF of 3 and 15 is 3. Divide the numerator and the denominator by 3. \(\frac{3 ÷ 3}{15 ÷ 3}\) = \(\frac{1}{3}\)

Question 7. \(\frac{3}{8}\) of the pets in the pet show are dogs. \(\frac{2}{3}\) of the dogs have long hair. What fraction of the pets are dogs with long hair? \(\frac{□}{□}\)

Answer: \(\frac{1}{4}\) are dogs with long hair

Explanation: \(\frac{3}{8}\) of the pets in the pet show are dogs. \(\frac{2}{3}\) of the dogs have long hair. \(\frac{3}{8}\) of \(\frac{2}{3}\) = \(\frac{3 × 2}{8 × 3}\) = \(\frac{6}{24}\) The GCF of 6 and 24 is 6. Divide the numerator and the denominator by 6. \(\frac{6 ÷ 6}{24 ÷ 6}\) = \(\frac{1}{4}\) \(\frac{1}{4}\) are dogs with long hair

Evaluate using the order of operations.

Question 8. \(\left(\frac{1}{2}+\frac{3}{8}\right) \times 8\) ______

Explanation: 1/2 + 3/8 = 7/8 7/8 × 8 = 7

Question 9. \(\frac{3}{4} \times\left(1-\frac{1}{9}\right)\) \(\frac{□}{□}\)

Answer: \(\frac{2}{3}\)

Explanation: 1 – 1/9 = 8/9 3/4 × 8/9 = 24/36 The GCF of 24 and 36 is 12. Divide the numerator and the denominator by 12. \(\frac{24 ÷ 12}{36 ÷ 12}\) = \(\frac{2}{3}\)

Question 10. \(4 \times \frac{1}{8} \times \frac{3}{10}\) \(\frac{□}{□}\)

Explanation: Multiply the numerators and Multiply the denominators. \(\frac{4 × 1 × 3}{1 × 8 × 10}\) = \(\frac{12}{80}\) Simplify using the GCF. The GCF of 12 and 80 is 4. Divide the numerator and the denominator by 4. \(\frac{12 ÷ 4}{80 ÷ 4}\) = \(\frac{3}{20}\)

Question 11. \(6 \times\left(\frac{4}{5}+\frac{2}{10}\right) \times \frac{2}{3}\) ______

Explanation: 4/5 + 2/10 = 1 6 × 1 × 2/3 = 12/3 The GCF of 12 and 3 is 4. Divide the numerator and the denominator by 3. \(\frac{12 ÷ 3}{3 ÷ 3}\) = \(\frac{4}{1}\) = 4

Question 12. Jason ran \(\frac{5}{7}\) of the distance around the school track. Sara ran \(\frac{4}{5}\) of Jason’s distance. What fraction of the total distance around the track did Sara run? \(\frac{□}{□}\)

Answer: \(\frac{4}{7}\)

Explanation: Jason ran \(\frac{5}{7}\) of the distance around the school track. Sara ran \(\frac{4}{5}\) of Jason’s distance. \(\frac{5}{7}\) × \(\frac{4}{5}\) = 20/35 The GCF of 20 and 35 is 5. Divide the numerator and the denominator by 5. \(\frac{20 ÷ 5}{35 ÷ 5}\) = \(\frac{4}{7}\)

Question 13. A group of students attend a math club. Half of the students are boys and \(\frac{4}{9}\) of the boys have brown eyes. What fraction of the group are boys with brown eyes? \(\frac{□}{□}\)

Answer: \(\frac{2}{9}\) group are boys with brown eyes

Explanation: A group of students attend a math club. Half of the students are boys and \(\frac{4}{9}\) of the boys have brown eyes. \(\frac{4}{9}\) × \(\frac{1}{2}\) = 4/18 = 2/9 2/9 group are boys with brown eyes

Question 14. Write and solve a word problem that involves multiplying by a fraction. Type below: __________

Answer: A group of students attends a math club. Half of the students are boys and \(\frac{6}{9}\) of the boys have brown eyes. What fraction of the group are boys with brown eyes? \(\frac{□}{□}\) Answer: A group of students attends a math club. Half of the students are boys and \(\frac{6}{9}\) of the boys have brown eyes. \(\frac{6}{9}\) × \(\frac{1}{2}\) = 6/18 = 1/3 1/3 group are boys with brown eyes.

Lesson Check – Page No. 86

Question 1. Veronica’s mom left \(\frac{3}{4}\) of a cake on the table. Her brothers ate \(\frac{1}{2}\) of it. What fraction of the cake did they eat? \(\frac{□}{□}\)

Answer: \(\frac{2}{4}\)

Explanation: Veronica’s mom left \(\frac{3}{4}\) of a cake on the table. Her brothers ate \(\frac{1}{2}\) of it. Since the fraction of the eaten cake is 1/2, you can multiply the numerator and denominator by and get an equivalent fraction, which is 2/4.

Question 2. One lap around the school track is \(\frac{5}{8}\) mile. Carin ran 3 \(\frac{1}{2}\) laps. How far did she run? _____ \(\frac{□}{□}\)

Answer: 2\(\frac{3}{16}\)

Explanation: One lap around the school track is \(\frac{5}{8}\) mile. Carin ran 3 \(\frac{1}{2}\) laps. 3 \(\frac{1}{2}\) = \(\frac{7}{2}\) Therefore, the total distance covered = 7/2 × 5/8 = 35/16 = 2 3/16

Question 3. Tom bought 2 \(\frac{5}{16}\) pounds of peanuts and 2.45 pounds of cashews. Which did he buy more of? Explain. Type below: __________

Explanation: Tom bought 2 \(\frac{5}{16}\) pounds of peanuts and 2.45 pounds of cashews. 2 \(\frac{5}{16}\) = 2.3125 2.3125 < 2.45 He buys more cashews.

Question 4. Eve has 24 stamps each valued at $24.75. What is the total value of her stamps? $ _____

Answer: $594

Explanation: Eve has 24 stamps each valued at $24.75. 24 x $24.75 = $594

Question 5. Naomi went on a 6.5-mile hike. In the morning, she hiked 1.75 miles, rested, and then hiked 2.4 more miles. She completed the hike in the afternoon. How much farther did she hike in the morning than in the afternoon? _____ miles

Answer: Naomi went on a 6.5-mile hike. In the morning, she hiked 1.75 miles, rested, and then hiked 2.4 more miles. She completed the hike in the afternoon. To find how many miles she walked in the afternoon you just subtract the morning miles 4.15 from the total miles 6.5. 6.5 – 4.15  = 2.35 To find how many more miles she walked in the morning you just subtract the morning from the afternoon 4.15 – 2.35=1.8 miles. She hiked 1.8 more miles in the morning

Question 6. A bookstore owner has 48 science fiction books and 30 mysteries he wants to sell quickly. He will make discount packages with one type of book in each. He wants the most books possible in each package, but all packages must contain the same number of books. How many packages can he make? How many packages of each type of book does he have? Type below: __________

Answer: 18 packages

Explanation: The bookstore owner can make 18 possible packages 48 – 30 = 18 packages

Share and Show – Page No. 89

Find the product. Simplify before multiplying.

Question 1. \(\frac{5}{6} \times \frac{3}{10}\) \(\frac{□}{□}\)

Answer: \(\frac{1}{4}\)

Explanation: \(\frac{5}{6} \times \frac{3}{10}\) Multiply the numerators and Multiply the denominators. \(\frac{5 × 3}{6 × 10}\) = \(\frac{15}{60}\) Simplify using the GCF. The GCF of 15 and 60 is 15. Divide the numerator and the denominator by 15. \(\frac{15 ÷ 15}{60 ÷ 15}\) = \(\frac{1}{4}\)

Question 2. \(\frac{3}{4} \times \frac{5}{9}\) \(\frac{□}{□}\)

Answer: \(\frac{5}{12}\)

Explanation: \(\frac{3}{4} \times \frac{5}{9}\) Multiply the numerators and Multiply the denominators. \(\frac{3 × 5}{4 × 9}\) = \(\frac{15}{36}\) Simplify using the GCF. The GCF of 15 and 36 is 3. Divide the numerator and the denominator by 3. \(\frac{15 ÷ 3}{36 ÷ 3}\) = \(\frac{5}{12}\)

Question 3. \(\frac{2}{3} \times \frac{9}{10}\) \(\frac{□}{□}\)

Explanation: \(\frac{2}{3} \times \frac{9}{10}\) Multiply the numerators and Multiply the denominators. \(\frac{2 × 9}{3 × 10}\) = \(\frac{18}{30}\) Simplify using the GCF. The GCF of 18 and 30 is 6. Divide the numerator and the denominator by 6. \(\frac{18 ÷ 6}{30 ÷ 6}\) = \(\frac{3}{5}\)

Question 4. After a picnic, \(\frac{5}{12}\) of the cornbread is left over. Val eats \(\frac{3}{5}\) of the leftover cornbread. What fraction of the cornbread does Val eat? \(\frac{□}{□}\)

Explanation: After a picnic, \(\frac{5}{12}\) of the cornbread is left over. Val eats \(\frac{3}{5}\) of the leftover cornbread. \(\frac{5}{12} \times \frac{3}{5}\) Multiply the numerators and Multiply the denominators. \(\frac{5 × 3}{12 × 5}\) = \(\frac{15}{60}\) Simplify using the GCF. The GCF of 15 and 60 is 15. Divide the numerator and the denominator by 15. \(\frac{15 ÷ 15}{60 ÷ 15}\) = \(\frac{1}{4}\)

Question 5. The reptile house at the zoo has an iguana that is \(\frac{5}{6}\) yd long. It has a Gila monster that is \(\frac{4}{5}\) of the length of the iguana. How long is the Gila monster? \(\frac{□}{□}\)

Explanation: The reptile house at the zoo has an iguana that is \(\frac{5}{6}\) yd long. It has a Gila monster that is \(\frac{4}{5}\) of the length of the iguana. \(\frac{5}{6} \times \frac{4}{5}\) Multiply the numerators and Multiply the denominators. \(\frac{5 × 4}{6× 5}\) = \(\frac{20}{30}\) Simplify using the GCF. The GCF of 20 and 30 is 10. Divide the numerator and the denominator by 10. \(\frac{20 ÷ 10}{30 ÷ 10}\) = \(\frac{2}{3}\)

Question 6. \(\frac{3}{4} \times \frac{1}{6}\) \(\frac{□}{□}\)

Explanation: \(\frac{3}{4} \times \frac{1}{6}\) Multiply the numerators and Multiply the denominators. \(\frac{3 × 1}{4 × 6}\) = \(\frac{3}{24}\) Simplify using the GCF. The GCF of 3 and 24 is 3. Divide the numerator and the denominator by 3. \(\frac{3 ÷ 3}{24 ÷ 3}\) = \(\frac{1}{8}\)

Question 7. \(\frac{7}{10} \times \frac{2}{3}\) \(\frac{□}{□}\)

Answer: \(\frac{7}{15}\)

Explanation: \(\frac{7}{10} \times \frac{2}{3}\) Multiply the numerators and Multiply the denominators. \(\frac{7 × 2}{10 × 3}\) = \(\frac{14}{30}\) Simplify using the GCF. The GCF of 14 and 30 is 2. Divide the numerator and the denominator by 2. \(\frac{14 ÷ 2}{30 ÷ 2}\) = \(\frac{7}{15}\)

Question 8. \(\frac{5}{8} \times \frac{2}{5}\) \(\frac{□}{□}\)

Explanation: \(\frac{5}{8} \times \frac{2}{5}\) Multiply the numerators and Multiply the denominators. \(\frac{5 × 2}{8 × 5}\) = \(\frac{10}{40}\) Simplify using the GCF. The GCF of 10 and 40 is 10. Divide the numerator and the denominator by 10. \(\frac{10 ÷ 10}{40 ÷ 10}\) = \(\frac{1}{4}\)

Question 9. \(\frac{9}{10} \times \frac{5}{6}\) \(\frac{□}{□}\)

Answer: \(\frac{3}{4}\)

Explanation: \(\frac{9}{10} \times \frac{5}{6}\) Multiply the numerators and Multiply the denominators. \(\frac{9 × 5}{10 × 6}\) = \(\frac{45}{60}\) Simplify using the GCF. The GCF of 45 and 60 is 15. Divide the numerator and the denominator by 15. \(\frac{45 ÷ 15}{60 ÷ 15}\) = \(\frac{3}{4}\)

Question 10. \(\frac{11}{12} \times \frac{3}{7}\) \(\frac{□}{□}\)

Answer: \(\frac{11}{28}\)

Explanation: \(\frac{11}{12} \times \frac{3}{7}\) Multiply the numerators and Multiply the denominators. \(\frac{11 × 3}{12 × 7}\) = \(\frac{33}{84}\) Simplify using the GCF. The GCF of 33 and 84 is 3. Divide the numerator and the denominator by 3. \(\frac{33 ÷ 3}{84 ÷ 3}\) = \(\frac{11}{28}\)

Question 11. Shelley’s basketball team won \(\frac{3}{4}\) of their games last season. In \(\frac{1}{6}\) of the games they won, they outscored their opponents by more than 10 points. What fraction of their games did Shelley’s team win by more than 10 points? \(\frac{□}{□}\)

Explanation: Let the total number of games be x. Number of games Shelley’s team won = 3/4x The number of games they outscored their opponents by more than 10 points = 1/6 X 3/4x = 1/8x Hence, in 1/8 of the total games, Shelley’s team won by 10 points.

Question 12. Mr. Ortiz has \(\frac{3}{4}\) pound of oatmeal. He uses \(\frac{2}{3}\) of the oatmeal to bake muffins. How much oatmeal does Mr. Ortiz have left? \(\frac{□}{□}\)

Explanation: Mr. Ortiz has \(\frac{3}{4}\) pound of oatmeal. He uses \(\frac{2}{3}\) of the oatmeal to bake muffins. \(\frac{3}{4} \times \frac{2}{3}\) Multiply the numerators and Multiply the denominators. \(\frac{3 × 2}{4 × 3}\) = \(\frac{6}{12}\) Simplify using the GCF. The GCF of 6 and 12 is 6. Divide the numerator and the denominator by 6. \(\frac{6 ÷ 6}{12 ÷ 6}\) = \(\frac{1}{2}\)

Question 13. Compare Strategies To find \(\frac{16}{27}\) × \(\frac{3}{4}\), you can multiply the fractions and then simplify the product or you can simplify the fractions and then multiply. Which method do you prefer? Explain. Type below: __________

Answer: \(\frac{16}{27}\) × \(\frac{3}{4}\) \(\frac{16 × 3}{27 × 4}\) = \(\frac{16 × 3}{4 × 27}\) \(\frac{48}{96}\) Simplify using the GCF. The GCF of 48 and 96 is 48. Divide the numerator and the denominator by 48. \(\frac{48 ÷ 48}{96 ÷ 48}\) = \(\frac{1}{2}\)

Problem Solving + Applications – Page No. 90

Question 14. Three students each popped \(\frac{3}{4}\) cup of popcorn kernels. The table shows the fraction of each student’s kernels that did not pop. Which student had \(\frac{1}{16}\) cup unpopped kernels? __________

Answer: Mirza

Explanation: Three students each popped \(\frac{3}{4}\) cup of popcorn kernels. The table shows the fraction of each student’s kernels that did not pop. Katie = 3/4 x 1/10 = 3/40 Mirza = 3/4 x 1/12 = 1/16

Question 15. The jogging track at Francine’s school is \(\frac{3}{4}\) mile long. Yesterday Francine completed two laps on the track. If she ran \(\frac{1}{3}\) of the distance and walked the remainder of the way, how far did she walk? ____ mile

Answer: 1 mile

Explanation: Length of jogging track at Francine’s school = 3/4 mile Let the distance covered by running be = x Let the distance covered by walking be = y Total number of laps completed by Francine = 2 Total distance covered by Francine = number of laps X distance covered in one lap 2 x 3/4 = 3/25 mile Now, distance covered by running = 1/3 of the total distance x = 1/3 x 3/2 distance covered by walking y = total distance – distance covered by running 3/2 – x = 3/2 – 1/2 = 1 mile Hence, Francine walked for 1 mile.

Question 16. At a snack store, \(\frac{7}{12}\) of the customers bought pretzels and \(\frac{3}{10}\) of those customers bought low-salt pretzels. Bill states that \(\frac{7}{30}\) of the customers bought low-salt pretzels. Does Bill’s statement make sense? Explain. Type below: __________

Answer: Bill’s statement does not make sense because it is incorrect: 7/12 customers bought pretzels. 3/10 Of those customers bought low-salt pretzels (x) 3/10 of 7/12 = x 21/120 = x Simplify: 7/40 To be correct, Bill would have to say that 7/40 of the customers bought low-salt pretzels, but instead, he had said 7/30.

Simplify Factors – Page No. 91

Question 1. \(\frac{8}{9} \times \frac{5}{12}\) \(\frac{□}{□}\)

Answer: \(\frac{10}{27}\)

Explanation: \(\frac{8}{9} \times \frac{5}{12}\) Multiply the numerators and Multiply the denominators. \(\frac{8 × 5}{9 × 12}\) = \(\frac{40}{108}\) Simplify using the GCF. The GCF of 40 and 108 is 4. Divide the numerator and the denominator by 4. \(\frac{40 ÷ 4}{108 ÷ 4}\) = \(\frac{10}{27}\)

Question 2. \(\frac{3}{4} \times \frac{16}{21}\) \(\frac{□}{□}\)

Explanation: \(\frac{3}{4} \times \frac{16}{21}\) Multiply the numerators and Multiply the denominators. \(\frac{3 × 16}{4 × 21}\) = \(\frac{48}{84}\) Simplify using the GCF. The GCF of 48 and 84 is 12. Divide the numerator and the denominator by 12. \(\frac{48 ÷ 12}{84 ÷ 12}\) = \(\frac{4}{7}\)

Question 3. \(\frac{15}{20} \times \frac{2}{5}\) \(\frac{□}{□}\)

Explanation: \(\frac{15}{20} \times \frac{2}{5}\) Multiply the numerators and Multiply the denominators. \(\frac{15 × 2}{20 × 5}\) = \(\frac{30}{100}\) Simplify using the GCF. The GCF of 30 and 100 is 10. Divide the numerator and the denominator by 10. \(\frac{30 ÷ 10}{100 ÷ 10}\) = \(\frac{3}{10}\)

Question 4. \(\frac{9}{18} \times \frac{2}{3}\) \(\frac{□}{□}\)

Explanation: \(\frac{9}{18} \times \frac{2}{3}\) Multiply the numerators and Multiply the denominators. \(\frac{9 × 2}{18 × 3}\) = \(\frac{18}{54}\) Simplify using the GCF. The GCF of 18 and 54 is 18. Divide the numerator and the denominator by 18. \(\frac{18 ÷ 18}{54 ÷ 18}\) = \(\frac{1}{3}\)

Question 5. \(\frac{3}{4} \times \frac{7}{30}\) \(\frac{□}{□}\)

Answer: \(\frac{7}{40}\)

Explanation: \(\frac{3}{4} \times \frac{7}{30}\) Multiply the numerators and Multiply the denominators. \(\frac{3 × 7}{4 × 30}\) = \(\frac{21}{120}\) Simplify using the GCF. The GCF of 21 and 120 is 3. Divide the numerator and the denominator by 3. \(\frac{21 ÷ 3}{120 ÷ 3}\) = \(\frac{7}{40}\)

Question 6. \(\frac{8}{15} \times \frac{15}{32}\) \(\frac{□}{□}\)

Explanation: \(\frac{8}{15} \times \frac{15}{32}\) Multiply the numerators and Multiply the denominators. \(\frac{8 × 15}{15 × 32}\) = \(\frac{120}{480}\) Simplify using the GCF. The GCF of 120 and 480 is 120. Divide the numerator and the denominator by 120. \(\frac{120 ÷ 120}{480 ÷ 120}\) = \(\frac{1}{4}\)

Question 7. \(\frac{12}{21} \times \frac{7}{9}\) \(\frac{□}{□}\)

Answer: \(\frac{4}{9}\)

Explanation: \(\frac{12}{21} \times \frac{7}{9}\) Multiply the numerators and Multiply the denominators. \(\frac{12 × 7}{21 × 9}\) = \(\frac{84}{189}\) Simplify using the GCF. The GCF of 84 and 189 is 21. Divide the numerator and the denominator by 21. \(\frac{84 ÷ 21}{189 ÷ 21}\) = \(\frac{4}{9}\)

Question 8. \(\frac{18}{22} \times \frac{8}{9}\) \(\frac{□}{□}\)

Answer: \(\frac{8}{11}\)

Explanation: \(\frac{18}{22} \times \frac{8}{9}\) Multiply the numerators and Multiply the denominators. \(\frac{18 × 8}{22 × 9}\) = \(\frac{144}{198}\) Simplify using the GCF. The GCF of 144 and 198 is 18. Divide the numerator and the denominator by 18. \(\frac{144 ÷ 18}{198 ÷ 18}\) = \(\frac{8}{11}\)

Question 9. Amber has a \(\frac{4}{5}\)-pound bag of colored sand. She uses \(\frac{1}{2}\) of the bag for an art project. How much sand does she use for the project? \(\frac{□}{□}\) pounds

Answer: \(\frac{2}{5}\) pounds

Explanation: Amber has a \(\frac{4}{5}\)-pound bag of colored sand. She uses \(\frac{1}{2}\) of the bag for an art project. 4/5 X 1/2 = 2/5

Question 10. Tyler has \(\frac{3}{4}\) month to write a book report. He finished the report in \(\frac{2}{3}\) at that time. How much time did it take Tyler to write the report? \(\frac{□}{□}\) month

Answer: \(\frac{1}{2}\) month

Explanation: Tyler has \(\frac{3}{4}\) month to write a book report. He finished the report in \(\frac{2}{3}\) at that time. 3/4 X 2/3 = 1/2

Question 11. Show two ways to multiply \(\frac{2}{15} \times \frac{3}{20}\). Then tell which way is easier and justify your choice. Type below: __________

Answer: \(\frac{2}{15} \times \frac{3}{20}\) 2/15 X 3/20 = 2/20 X 3/15 = 1/10 X 1/5 = 1/50

Lesson Check – Page No. 92

Find each product. Simplify before multiplying.

Question 1. At Susie’s school, \(\frac{5}{8}\) of all students play sports. Of the students who play sports, \(\frac{2}{5}\) play soccer. What fraction of the students in Susie’s school play soccer? \(\frac{□}{□}\)

Explanation: At Susie’s school, \(\frac{5}{8}\) of all students play sports. Of the students who play sports, \(\frac{2}{5}\) play soccer. Multiply 5/8 X 2/5, and the answer is 0.25, which converts to 25/100 or 1/4

Question 2. A box of popcorn weighs \(\frac{15}{16}\) pounds. The box contains \(\frac{1}{3}\) buttered popcorn and \(\frac{2}{3}\) cheesy popcorn. How much does the cheesy popcorn weigh? \(\frac{□}{□}\)

Answer: \(\frac{5}{8}\)

Explanation: Total weight of a box of popcorn = 15/16 pounds. We are given two types of popcorn are there, butter popcorn and cheesy popcorn. Butter popcorn is one-third of the total weight = 1/3 of the Total weight Plugging the value of the total weight, we get = 1/3 * 15/16 = 5/16 pounds. Cheesy popcorn = 2/3 of Total weight Plugging the value of the total weight, we get = 2/3 * 15/16 = 10/16 or 5/8 pounds. Therefore, cheesy popcorn weighs is 5/8 pounds.

Question 3. Ramòn bought a dozen ears of corn for $1.80. What was the cost of each ear of corn? $ ______

Answer: $0.15

Explanation: Ramòn bought a dozen ears of corn for $1.80. So, for the cost of each ear of corn, $1.80/12 = $0.15

Question 4. A 1.8-ounce jar of cinnamon costs $4.05. What is the cost per ounce? $ ______

Answer: $2.25 per ounce

Explanation: If a 1.8-ounce jar costs $4.05, do $4.05 divided by 1.8. $4.05 / 1.8 = $2.25 per ounce.

Question 5. Rose bought \(\frac{7}{20}\) kilogram of ginger candy and 0.4 kilogram of cinnamon candy. Which did she buy more of? Explain how you know. Type below: __________

Answer: Rose bought ginger candy = 7/20 kilogram = 0.35 Kilogram She bought cinnamon candy = 0.4 kilogram 0.4 > 0.35 Therefore, She bought cinnamon candy.

Question 6. Don walked 3 \(\frac{3}{5}\) miles on Friday, 3.7 miles on Saturday, and 3 \(\frac{5}{8}\) miles on Sunday. List the distances from least to greatest. Type below: __________

Answer: 3 \(\frac{3}{5}\), 3 \(\frac{5}{8}\), 3.7

Explanation: 3 \(\frac{3}{5}\) = 18/5 = 3.6 3 \(\frac{5}{8}\) = 29/8 = 3.625 3.6 < 3.625 < 3.7 3 \(\frac{3}{5}\), 3 \(\frac{5}{8}\), 3.7

Mid-Chapter Checkpoint – Vocabulary – Page No. 93

Question 1. The fractions \(\frac{1}{2}\) and \(\frac{5}{10}\) are _____. Type below: __________

Answer: Equivalent fractions

Question 2. A _____ is a denominator that is the same in two or more fractions. Type below: __________

Answer: Common Denominator

Concepts and Skills

Write as a decimal. Tell whether you used division, a number line, or some other method.

Question 3. \(\frac{7}{20}\) _____

Answer: 0.35

Explanation: By using Division, \(\frac{7}{20}\) = 0.35

Question 4. 8 \(\frac{39}{40}\) _____

Answer: 8.975

Explanation: By using Division, 8 \(\frac{39}{40}\) = 359/40 = 8.975

Question 5. 1 \(\frac{5}{8}\) _____

Answer: 1.625

Explanation: By using Division, 1 \(\frac{5}{8}\) = 13/8 = 1.625

Question 6. \(\frac{19}{25}\) _____

Answer: 0.76

Explanation: By using Division, \(\frac{19}{25}\) = 0.76

Question 7. \(\frac{4}{5}, \frac{3}{4}, 0.88\) Type below: __________

Answer: \(\frac{3}{4}\), \(\frac{4}{5}\),0.88

Explanation: Write the decimal form of 4/5 = 0.8 Write the decimal form of 3/4 = 0.75 0.88 0.75 < 0.8 < 0.88

Question 8. 0.65, 0.59, \(\frac{3}{5}\) Type below: __________

Answer: 0.59, \(\frac{3}{5}\), 0.65

Explanation: Write the decimal form of 3/5 = 0.6 0.59 < 0.6 < 0.65

Question 9. \(1 \frac{1}{4}, 1 \frac{2}{3}, \frac{11}{12}\) Type below: __________

Answer: \(\frac{11}{12}\), 1\(\frac{1}{4}\), 1\(\frac{2}{3}\)

Explanation: Write the decimal form of 1 1/4 = 5/4 = 1.25 Write the decimal form of 1 2/3 = 5/3 = 1.66 Write the decimal form of 11/12 = 0.916 0.916 < 1.25 < 1.66

Question 10. 0.9, \(\frac{7}{8}\), 0.86 Type below: __________

Answer: 0.86, \(\frac{7}{8}\), 0.9

Explanation: Write the decimal form of \(\frac{7}{8}\) = 0.875 0.86 < 0.875 < 0.9

Question 11. \(\frac{2}{3} \times \frac{1}{8}\) \(\frac{□}{□}\)

Answer: \(\frac{1}{12}\)

Explanation: \(\frac{2}{3} \times \frac{1}{8}\) Multiply the numerators and Multiply the denominators. \(\frac{2 × 1}{3 × 8}\) = \(\frac{2}{24}\) Simplify using the GCF. The GCF of 2 and 24 is 2. Divide the numerator and the denominator by 2. \(\frac{2 ÷ 2}{24 ÷ 2}\) = \(\frac{1}{12}\)

Question 12. \(\frac{4}{5} \times \frac{2}{5}\) \(\frac{□}{□}\)

Answer: \(\frac{8}{25}\)

Explanation: \(\frac{4}{5} \times \frac{2}{5}\) Multiply the numerators and Multiply the denominators. \(\frac{4 × 2}{5 × 5}\) = \(\frac{8}{25}\)

Question 13. 12 × \(\frac{3}{4}\) _____

Explanation: 12 × \(\frac{3}{4}\) Multiply the numerators and Multiply the denominators. \(\frac{12 × 3}{1 × 4}\) = \(\frac{36}{4}\) = 9

Question 14. Mia climbs \(\frac{5}{8}\) of the height of the rock wall. Lee climbs \(\frac{4}{5}\) of Mia’s distance. What fraction of the wall does Lee climb? \(\frac{□}{□}\)

Explanation: find the LCM (least common denominator) for 5/8 and 4/5. 5/8= 25/40 and 4/5= 32/40. Subtract and you get 7/40.

Page No. 94

Question 15. In Zoe’s class, \(\frac{4}{5}\) of the students have pets. Of the students who have pets, \(\frac{1}{8}\) have rodents. What fraction of the students in Zoe’s class have pets that are rodents? What fraction of the students in Zoe’s class have pets that are not rodents? Type below: __________

Answer: \(\frac{1}{10}\) of the students in Zoe’s class have pets that are rodents \(\frac{7}{10}\) of the students in Zoe’s class have pets that are not rodents

Explanation: In Zoe’s class, \(\frac{4}{5}\) of the students have pets. Of the students who have pets, \(\frac{1}{8}\) have rodents. 4/5 X 1/8 = 1/10 4/5 – 1/10 = 7/10

Question 16. A recipe calls for 2 \(\frac{2}{3}\) cups of flour. Terell wants to make \(\frac{3}{4}\) of the recipe. How much flour should he use? _____ cups

Answer: 2 cups

Explanation: 2 \(\frac{2}{3}\) = 8/3 8/3 * 3/4 = 2

Question 17. Following the Baltimore Running Festival in 2009, volunteers collected and recycled 3.75 tons of trash. Graph 3.75 on a number line and write the weight as a mixed number. Type below: __________

Answer: Volunteers collected and recycled 3.75 tons of trash. We need to convert 3.75 as a mixed number. The mixed number consists of a whole number and a proper fraction. In the given number 3.75, 3 as the whole number and convert 0.75 to a fraction. 3.75 = 3 + 0.75 = 3 + 75/100 We can reduce the fraction 75/ 100 = 3+ 3/4 = 3 3/4

Answer: 22/25 = 0.88 17/20 = 0.85 4/5 = 0.8 3/4 = 0.75 Monica had the highest score Let x be the total number of points: (22/25 + 17/20 + 4/5 + 3/4)x = 80 x = 24.39 That is not a whole number of points.

Share and Show – Page No. 97

Use the model to find the quotient.

Explanation: 1/2 groups of 3 \(\frac{1}{2}\) ÷ 3 1/2 × 1/3 = 1/6

Explanation: 3/4 groups of 3/8 3/4 × 8/3 = 2

Use fraction strips to find the quotient. Then draw the model.

Question 3. \(\frac{1}{3}\) ÷ 4 \(\frac{□}{□}\)

Explanation: \(\frac{1}{3}\) ÷ 4 \(\frac{1}{3}\) × \(\frac{1}{4}\) \(\frac{1}{12}\)

Question 4. \(\frac{3}{5} \div \frac{3}{10}\) ______

Explanation: \(\frac{3}{5} \div \frac{3}{10}\) \(\frac{3}{5}\) × \(\frac{10}{3}\) 2

Draw a model to solve. Then write an equation for the model. Interpret the result.

Question 5. How many \(\frac{1}{4}\) cup servings of raisins are in \(\frac{3}{8}\) cup of raisins? Type below: __________

Answer: 1.5

Explanation: 3/8 × 1/4 = 1.5

Question 6. How many \(\frac{1}{3}\) lb bags of trail mix can Josh make from \(\frac{5}{6}\) lb of trail mix? Type below: __________

Explanation: Multiply 1/3 with 2 1/3 × 2 = 2/6. 2/6 can go into 5/6 twice so the answer is two bags.

Additional Practice 2.5 Compare Decimals Question 7. Pose a Problem Write and solve a problem for \(\frac{3}{4}\) ÷ 3 that represents how much in each of 3 groups. Type below: __________

Explanation: \(\frac{3}{4}\) ÷ 3 \(\frac{3}{4}\) × \(\frac{1}{3}\) = 1/4

Problem Solving + Applications – Page No. 98

The table shows the amount of each material that students in a sewing class need for one purse.

Question 8. Mrs. Brown has \(\frac{1}{3}\) yd of blue denim and \(\frac{1}{2}\) yd of black denim. How many purses can be made using denim as the main fabric? _____ purses

Answer: 5 purses

Explanation: Mrs. Brown has \(\frac{1}{3}\) yd of blue denim and \(\frac{1}{2}\) yd of black denim. 3 + 2 = 5

Question 9. One student brings \(\frac{1}{2}\) yd of ribbon. If 3 students receive an equal length of the ribbon, how much ribbon will each student receive? Will each of them have enough ribbon for a purse? Explain. Type below: __________

Answer: One student brings \(\frac{1}{2}\) yd of ribbon. If 3 students receive an equal length of the ribbon, \(\frac{1}{2}\) ÷ 3 1/2 × 1/3 = 1/6 They don’t have enough ribbon for a purse

Question 10. Make Arguments There was \(\frac{1}{2}\) yd of purple and pink striped fabric. Jessie said she could only make \(\frac{1}{24}\) of a purse using that fabric as the trim. Is she correct? Use what you know about the meanings of multiplication and division to defend your answer. Type below: __________

Answer: There was \(\frac{1}{2}\) yd of purple and pink striped fabric. Jessie said she could only make \(\frac{1}{24}\) of a purse using that fabric as the trim. 1/2 × 12 = 1/24 So, 12 is the answer

Question 11. Draw a model to find the quotient. \(\frac{1}{2}\) ÷ 4 = Type below: __________

Explanation: 1/2 × 1/4 = 1/8

Model Fraction Division – Page No. 99

Use the model to find the quotient

Explanation: \(\frac{1}{4}\) ÷ 3 \(\frac{1}{4}\) × \(\frac{1}{3}\) = \(\frac{1}{12}\)

Explanation: \(\frac{1}{2} \div \frac{2}{12}=\) \(\frac{1}{2}\) × \(\frac{12}{2}\) = \(\frac{12}{4}\) = 3

Use fraction strips to find the quotient.

Question 3. \(\frac{5}{6} \div \frac{1}{2}=\) ______ \(\frac{□}{□}\)

Answer: \(\frac{5}{3}\)

Explanation: \(\frac{5}{6} \div \frac{1}{2}=\) \(\frac{5}{6}\) × \(\frac{2}{1}\) = \(\frac{5}{3}\)

Question 4. \(\frac{2}{3}\) ÷ 4 = \(\frac{□}{□}\)

Explanation: \(\frac{2}{3}\) ÷ 4 \(\frac{2}{3}\) × \(\frac{1}{4}\) = \(\frac{2}{12}\) = 1/6

Question 5. \(\frac{1}{2}\) ÷ 6 = \(\frac{□}{□}\)

Explanation: \(\frac{1}{2}\) ÷ 6 \(\frac{1}{2}\) × \(\frac{1}{6}\) = \(\frac{1}{12}\)

Question 6. \(\frac{1}{3} \div \frac{1}{12}\) ______

Explanation: \(\frac{1}{3} \div \frac{1}{12}\) \(\frac{1}{3}\) × \(\frac{12}{1}\) = \(\frac{12}{3}\) = 4

Question 7. If Jerry runs \(\frac{1}{10}\) mile each day, how many days will it take for him to run \(\frac{4}{5}\) mile? ______ days

Answer: 8 days

Explanation: If Jerry runs \(\frac{1}{10}\) mile each day, \(\frac{4}{5}\) ÷ \(\frac{1}{10}\) \(\frac{4}{5}\) × \(\frac{10}{1}\) = \(\frac{40}{5}\) = 8

Question 8. Mrs. Jennings has \(\frac{3}{4}\) gallon of paint for an art project. She plans to divide the paint equally into jars. If she puts \(\frac{1}{8}\) gallon of paint into each jar, how many jars will she use? ______ jars

Answer: 6 jars

Explanation: Mrs. Jennings has 3/4 Gallons of paint for an art project. In 1 jar she puts 1/8 gallon of paint. The number of jars in which she plans to divide the paint equally is given by, n= 3/4 ÷ 1/8 n = \(\frac{3}{4}\) × \(\frac{8}{1}\) = \(\frac{24}{4}\) = 6

Question 9. If one jar of glue weighs \(\frac{1}{12}\) pound, how many jars can Rickie get from \(\frac{2}{3}\) pound of glue? ______ jars

Answer: 8 jars

Explanation: The weight of glue in one jar = 1/12 pound To get 2/3 pound of glue Rickie can get the number of jars 2/3 ÷ 1/12 2/3 × 12/1 = 24/3 = 8

Question 10. Explain how to use a model to show \(\frac{2}{6} \div \frac{1}{12}\) and \(\frac{2}{6}\) ÷ 4. Type below: __________

Explanation: \(\frac{2}{6} \div \frac{1}{12}\) 2/6 = 1/3 1/3 x 12/1 = 4 \(\frac{2}{6}\) ÷ 4 1/3 x 1/4 = 1/12

Lesson Check – Page No. 100

Question 1. Darcy needs \(\frac{1}{4}\) yard of fabric to make a banner. She has 2 yards of fabric. How many banners can she make? ______ banners

Answer: 8 banners

Explanation: Darcy needs \(\frac{1}{4}\) yard of fabric to make a banner. She has 2 yards of fabric. 2 ÷ \(\frac{1}{4}\) = 2 x 4 = 8

Question 2. Lorenzo bought \(\frac{15}{16}\) pounds of ground beef. He wants to make hamburgers that weigh \(\frac{3}{16}\) pound each. How many hamburgers can he make? ______ hamburgers

Answer: 5 hamburgers

Explanation: Lorenzo bought \(\frac{15}{16}\) pounds of ground beef. He wants to make hamburgers that weigh \(\frac{3}{16}\) pound each. \(\frac{15}{16}\) ÷ \(\frac{3}{16}\) 15/3 = 5

Question 3. Letisha wants to read 22 pages a night. At that rate, how long will it take her to read a book with 300 pages? ______ nights

Answer: 14 nights

Explanation: Letisha wants to read 22 pages a night. It takes her to read a book with 300 pages 300/22 = 13.6 13.6 is near to 14 So, it is for 2 weeks.

Question 4. A principal wants to order enough notebooks for 624 students. The notebooks come in boxes of 28. How many boxes should he order? ______ boxes

Answer: 22 boxes

Explanation: A principal wants to order enough notebooks for 624 students. The notebooks come in boxes of 28. 624/28 = 22.2857 22.2857 is closer to 22 22 boxes.

Question 5. Each block in Ton’s neighborhood is \(\frac{2}{3}\) mile long. If he walks 4 \(\frac{1}{2}\) blocks, how far does he walk? ______ miles

Answer: 3 miles

Explanation: If each block is 2/3 miles long, and he walks 4 1/2 blocks, we can simply multiply to two. It looks like this: (2/3)(4 1/2) to multiply, make 4 1/2 into an improper fraction and multiply normally (2/3)(9/4) Ton walks 3 miles total.

Question 6. In Cathy’s garden, \(\frac{5}{6}\) of the area is planted with flowers. Of the flowers, \(\frac{3}{10}\) of them are red. What fraction of Cathy’s garden is planted with red flowers? \(\frac{□}{□}\)

Explanation: In Cathy’s garden, \(\frac{5}{6}\) of the area is planted with flowers. Of the flowers, \(\frac{3}{10}\) of them are red. 5/6 x 3/10 = 1/4

Share and Show – Page No. 103

Estimate using compatible numbers.

Question 1. \(22 \frac{4}{5} \div 6 \frac{1}{4}\) _______

Explanation: 22 \(\frac{4}{5}\) = 114/5 = 22.8 6 \(\frac{1}{4}\) = 25/4 = 6.25 22.8 is closer to 24 6.25 is closer to 6 24/6 = 4

Question 2. \(12 \div 3 \frac{3}{4}\) _______

Explanation: 3 \(\frac{3}{4}\) = 15/4 = 3.75 3.75 is closer to 4 12/4 = 3

Question 3. \(33 \frac{7}{8} \div 5 \frac{1}{3}\) _______

Explanation: 33 \(\frac{7}{8}\) = 271/8 = 33.875 5 \(\frac{1}{3}\) = 16/3 = 5.333 33.875 is closer to 35 5.333 is closer to 5 35/5 = 7

Question 4. \(3 \frac{7}{8} \div \frac{5}{9}\) _______

Explanation: 3 \(\frac{7}{8}\) = 31/8 = 3.875 \(\frac{5}{9}\) = 0.555 3.875 is closer to 4 0.555 is closer to 1 4/1 = 4

Additional Practice 2.6 Round Decimals Answer Key Question 5. \(34 \frac{7}{12} \div 7 \frac{3}{8}\) _______

Explanation: 34 \(\frac{7}{12}\) = 415/12 = 34.583 7 \(\frac{3}{8}\) = 59/8 = 7.375 34.583 is closer to 35 7.375 is closer to 7 35/7 = 5

Question 6. \(1 \frac{2}{9} \div \frac{1}{6}\) _______

Explanation: 1 \(\frac{2}{9}\) = 11/9 = 1.222 \(\frac{1}{6}\) = 0.1666 1.222 is closer to 1 0.1666 is closer to 0.2 1/0.2 = 5

Question 7. \(44 \frac{1}{4} \div 11 \frac{7}{9}\) _______

Explanation: 44 \(\frac{1}{4}\) = 177/4 = 44.25 11 \(\frac{7}{9}\) = 106/9 = 11.77 44.25 is closer to 44 11.77 is closer to 11 44/11 = 4

Question 8. \(71 \frac{11}{12} \div 8 \frac{3}{4}\) _______

Explanation: 71 \(\frac{11}{12}\) = 863/12 = 71.916 8 \(\frac{3}{4}\) = 35/4 = 8.75 71.916 is closer to 72 8.75 is closer to 9 72/9 = 8

Question 9. \(1 \frac{1}{6} \div \frac{1}{8}\) _______

Explanation: 1 \(\frac{1}{6}\) = 7/6 = 1.166 \(\frac{1}{8}\) = 0.125 1.166 is closer to 1.2 0.125 is closer to 0.1 1.2/0.1 = 12

Estimate to compare. Write <, >, or =.

Question 10. \(21 \frac{3}{10} \div 2 \frac{5}{6}\) _______ \(35 \frac{7}{9} \div 3 \frac{2}{3}\)

Answer: \(21 \frac{3}{10} \div 2 \frac{5}{6}\) < \(35 \frac{7}{9} \div 3 \frac{2}{3}\)

Explanation: 21 \(\frac{3}{10}\) = 213/10 = 21.3 2 \(\frac{5}{6}\) = 17/6 = 2.833 21.3 is closer to 21 2.833 is closer to 3 21/3 = 7 35 \(\frac{7}{9}\) = 322/9 = 35.777 3 \(\frac{2}{3}\) = 11/3 = 3.666 35.777 is closer to 36 3.666 is closer to 4 36/4 = 9 7 < 9 So, \(21 \frac{3}{10} \div 2 \frac{5}{6}\) < \(35 \frac{7}{9} \div 3 \frac{2}{3}\)

Question 11. \(29 \frac{4}{5} \div 5 \frac{1}{6}\) _______ \(27 \frac{8}{9} \div 6 \frac{5}{8}\)

Answer: \(29 \frac{4}{5} \div 5 \frac{1}{6}\) > \(27 \frac{8}{9} \div 6 \frac{5}{8}\)

Explanation: 29 \(\frac{4}{5}\) = 149/5 = 29.8 5 \(\frac{1}{6}\) = 31/6 = 5.1666 29.8 is closer to 30 5.1666 is closer to 5 30/5 = 6 27 \(\frac{8}{9}\) = 251/9 = 27.888 6 \(\frac{5}{8}\) = 53/8 = 6.625 27.888 is closer to 30 6.625 is closer 7 30/7 = 5 6 > 5 \(29 \frac{4}{5} \div 5 \frac{1}{6}\) > \(27 \frac{8}{9} \div 6 \frac{5}{8}\)

Question 12. \(55 \frac{5}{6} \div 6 \frac{7}{10}\) _______ \(11 \frac{5}{7} \div \frac{5}{8}\)

Answer: \(55 \frac{5}{6} \div 6 \frac{7}{10}\) < \(11 \frac{5}{7} \div \frac{5}{8}\)

Explanation: 55 \(\frac{5}{6}\) = 335/6 = 55.833 6 \(\frac{7}{10}\) = 67/10 = 6.7 55.833 is closer to 56 6.7 is closer to 7 56/7 = 8 11 \(\frac{5}{7}\) = 82/7 = 11.714 \(\frac{5}{8}\) = 0.625 11.714 is closer to 12 0.625 is closer to 1 12/1 = 12 8 < 12

Question 13. Marion is making school flags. Each flag uses 2 \(\frac{3}{4}\) yards of felt. Marion has 24 \(\frac{1}{8}\) yards of felt. About how many flags can he make? About _______ flags

Answer: About 8 flags

Explanation: Marion is making school flags. Each flag uses 2 \(\frac{3}{4}\) yards of felt. Marion has 24 \(\frac{1}{8}\) yards of felt. 2 \(\frac{3}{4}\) = 11/4 24 \(\frac{1}{8}\) = 193/8 193/8 ÷ 11/4 193/8 x 4/11 = 8.77 About 8 flags

Question 14. A garden snail travels about 2 \(\frac{3}{5}\) feet in 1 minute. At that speed, about how many hours would it take the snail to travel 350 feet? About _______ hours

Answer: About 2 hours

Explanation: 2 \(\frac{3}{5}\) = 2.6 That’s how long he travels in one minute. There are 60 minutes in an hour so multiply it by 60 and see if that gets you close to 350. 60 x 2.6 = 156 Now let’s add one more hour. 156 + 156 = 312 14 x 2.6 = 36.4 312 + 36.4 = 348.4 348.4 + 2.6 = 351 So two hours and fourteen minutes

Problem Solving + Applications – Page No. 104

What’s the Error?

Question 15. Megan is making pennants from a piece of butcher paper that is 10 \(\frac{3}{8}\) yards long. Each pennant requires \(\frac{3}{8}\) yard of paper. To estimate the number of pennants she could make, Megan estimated the quotient 10 \(\frac{3}{8}\) ÷ \(\frac{3}{8}\). Look at how Megan solved the problem. Find her error Estimate: 10 \(\frac{3}{8}\) ÷ \(\frac{3}{8}\) 10 ÷ \(\frac{1}{2}\) = 5 Correct the error. Estimate the quotient. So, Megan can make about _____ pennants. Describe the error that Megan made Explain Tell which compatible numbers you used to estimate 10 \(\frac{3}{8}\) ÷ \(\frac{3}{8}\). Explain why you chose those numbers. Type below: __________

Answer: 10 \(\frac{3}{8}\) ÷ \(\frac{3}{8}\) 10 \(\frac{3}{8}\) = 83/8 = 10.375 \(\frac{3}{8}\) = 0.375 She had written 10 ÷ \(\frac{1}{2}\) = 5 10.375 is closer to 10 0.375 is closer to 0.5 10/0.5 = 20 But she has written 5 instead of 20. Megan can make about 20 pennants.

For numbers 16a–16c, estimate to compare. Choose <, >, or =.

Question 16. 16a. 18 \(\frac{3}{10} \div 2 \frac{5}{6}\) ? \(30 \frac{7}{9} \div 3 \frac{1}{3}\) _____

Answer: 16a. 18 \(\frac{3}{10} \div 2 \frac{5}{6}\) < \(30 \frac{7}{9} \div 3 \frac{1}{3}\)

Explanation: 18 \(\frac{3}{10}\) = 183/10 = 18.3 2 \(\frac{5}{6}\) = 17/6 = 2.833 18.3 is closer to 18 2.833 is closer to 3 18/3 = 6 30 \(\frac{7}{9}\) = 277/9 = 30.777 3 \(\frac{1}{3}\) = 10/3 = 3.333 30.777 is closer to 30 3.333 is closer to 3 30/3 = 10 6 < 10

Question 16. 16b. 17 \(\frac{4}{5} \div 6 \frac{1}{6}\) ? \(19 \frac{8}{9} \div 4 \frac{5}{8}\) _____

Answer: 17 \(\frac{4}{5} \div 6 \frac{1}{6}\) < \(19 \frac{8}{9} \div 4 \frac{5}{8}\)

Explanation: 17 \(\frac{4}{5}\) = 89/5 = 17.8 6 \(\frac{1}{6}\) = 37/6 = 6.1666 17.8 is closer to 18 6.1666 is closer to 6 18/6 = 3 19 \(\frac{8}{9}\) = 179/9 = 19.888 4 \(\frac{5}{8}\) = 37/8 = 4.625 19.888 is closer to 20 4.625 is closer to 5 20/5 = 4 3 < 4 17 \(\frac{4}{5} \div 6 \frac{1}{6}\) < \(19 \frac{8}{9} \div 4 \frac{5}{8}\)

Question 16. 16c. 17 \(\frac{5}{6} \div 6 \frac{1}{4}\) ? \(11 \frac{5}{7} \div 2 \frac{3}{4}\) _____

Answer: 17 \(\frac{5}{6} \div 6 \frac{1}{4}\) < \(11 \frac{5}{7} \div 2 \frac{3}{4}\)

Explanation: 17 \(\frac{5}{6}\) = 107/6 = 17.833 6 \(\frac{1}{4}\) = 25/4 = 6.25 17.833 is closer to 18 6.25 is closer to 6 18/6 = 3 11 \(\frac{5}{7}\) = 82/7 = 11.714 2 \(\frac{3}{4}\) = 11/4 = 2.75 11.714 is closer to 12 2.75 is closer to 3 12/3 = 4 3 < 4 17 \(\frac{5}{6} \div 6 \frac{1}{4}\) < \(11 \frac{5}{7} \div 2 \frac{3}{4}\)

Estimate Quotients – Page No. 105

Question 1. \(12 \frac{3}{16} \div 3 \frac{9}{10}\) ______

Explanation: 12 \(\frac{3}{16}\) = 195/16 = 12.1875 3 \(\frac{9}{10}\) = 39/10 = 3.9 12.1875 is closer to 12 3.9 is closer to 4 12/4 = 3

Question 2. \(15 \frac{3}{8} \div \frac{1}{2}\) ______

Explanation: 15 \(\frac{3}{8}\) = 123/8 = 15.375 \(\frac{1}{2}\) = 0.5 15.375 is closer to 15 0.5 is closer to 0.5 15/0.5 = 30

Question 3. \(22 \frac{1}{5} \div 1 \frac{5}{6}\) ______

Explanation: 22 \(\frac{1}{5}\) = 111/5 = 22.2 1 \(\frac{5}{6}\) = 11/6 = 1.8333 22.2 is closer to 22 1.8333 is closer to 2 22/2 = 11

Question 4. \(7 \frac{7}{9} \div \frac{4}{7}\) ______

Explanation: 7 \(\frac{7}{9}\) = 70/9 = 7.777 \(\frac{4}{7}\) = 0.571 7.777 is closer to 8 0.571 is closer to 0.5 8/0.5 = 16

Question 5. \(18 \frac{1}{4} \div 2 \frac{4}{5}\) ______

Explanation: 18 \(\frac{1}{4}\) = 73/4 = 18.25 2 \(\frac{4}{5}\) = 14/5 = 2.8 18.25 is closer to 18 2.8 is closer to 3 18/3 = 6

Question 6. \(\frac{15}{16} \div \frac{1}{7}\) ______

Explanation: \(\frac{15}{16}\) = 0.9375 \(\frac{1}{7}\) = 0.1428 0.9375 is closer to 1 0.1428 is closer to 0.1 1/0.1 = 10

Question 7. \(14 \frac{7}{8} \div \frac{5}{11}\) ______

Explanation: 14 \(\frac{7}{8}\) = 119/8 = 14.875 \(\frac{5}{11}\) = 0.4545 14.875 is closer to 15 0.4545 is closer to 0.5 15/0.5 = 30

Question 8. \(53 \frac{7}{12} \div 8 \frac{11}{12}\) ______

Explanation: 53 \(\frac{7}{12}\) = 643/12 = 53.58 8 \(\frac{11}{12}\) = 107/12 = 8.916 53.58 is closer to 54 8.916 is closer to 9 54/9 = 6

Question 9. \(1 \frac{1}{6} \div \frac{1}{9}\) ______

Explanation: 1 \(\frac{1}{6}\) = 7/6 = 1.166 \(\frac{1}{9}\) = 0.111 1.166 is closer to 1 0.111 is closer to 0.1 1/0.1 = 10

Question 10. Estimate the number of pieces Sharon will have if she divides 15 \(\frac{1}{3}\) yards of fabric into 4 \(\frac{4}{5}\) yard lengths. About ______ pieces

Answer: About 3 pieces

Explanation: Sharon will have if she divides 15 \(\frac{1}{3}\) yards of fabric into 4 \(\frac{4}{5}\) yard lengths. 3 7/36 is the answer. So, about 3 pieces

Question 11. Estimate the number of \(\frac{1}{2}\) quart containers Ethan can fill from a container with 8 \(\frac{7}{8}\) quarts of water. About ______ containers

Answer: About 18 containers

Question 12. How is estimating quotients different from estimating products? Type below: __________

Answer: To estimate products and quotients, you need to first round the numbers. To round to the nearest whole number, look at the digit in the tenths place. If it is less than 5, round down. If it is 5 or greater, round up. Remember that an estimate is an answer that is not exact, but is approximate and reasonable. Let’s look at an example of estimating a product. Estimate the product: 11.256×6.81 First, round the first number. Since there is a 2 in the tenths place, 11.256 rounds down to 11. Next, round the second number. Since there is an 8 in the tenths place, 6.81 rounds up to 7. Then, multiply the rounded numbers. 11×7=77 The answer is 77. Let’s look at an example of estimating a quotient. Estimate the quotient: 91.93÷4.39 First, round the first number. Since there is a 9 in the tenths place, 91.93 rounds up to 92. Next, round the second number. Since there is a 3 in the tenths place, 4.39 rounds down to 4. Then, divide the rounded numbers. 92÷4=23 The answer is 23.

Lesson Check – Page No. 106

Question 1. Each loaf of pumpkin bread calls for 1 \(\frac{3}{4}\) cups of raisins. About how many loaves can be made from 10 cups of raisins? About ______ loaves

Answer: About 5 loaves

Explanation: Divide 10 by 1 3/4. The answer is 5.714285 So you can make about 5 loaves of bread with 10 cups of raisins if each loaf needs 1 3/4 cups of raisins.

Question 2. Perry’s goal is to run 2 \(\frac{1}{4}\) miles each day. One lap around the school track is \(\frac{1}{3}\) mile. About how many laps must he run to reach his goal? About ______ laps

Answer: About 9 laps

Explanation: Perry’s goal is to run 2 \(\frac{1}{4}\) miles each day. One lap around the school track is \(\frac{1}{3}\) mile. 2 \(\frac{1}{4}\) = 9/4 = 2.25 \(\frac{1}{3}\) = 0.333 Perry will have to run 9 laps to reach his goal.

Question 3. A recipe calls for \(\frac{3}{4}\) teaspoon of red pepper. Uri wants to use \(\frac{1}{3}\) of that amount. How much red pepper should he use? \(\frac{□}{□}\) teaspoon

Answer: \(\frac{1}{4}\) teaspoon

Explanation: A recipe calls for \(\frac{3}{4}\) teaspoon of red pepper. Uri wants to use \(\frac{1}{3}\) of that amount. \(\frac{1}{3}\) of \(\frac{3}{4}\) = \(\frac{1}{4}\)

Question 4. A recipe calls for 2 \(\frac{2}{3}\) cups of apple slices. Zoe wants to use 1 \(\frac{1}{2}\) times this amount. How many cups of apples should Zoe use? ______ cups

Answer: 4 cups

Explanation: A recipe calls for 2 2/3 cups of apple slices. Zoe wants to use 1 1/2 times this amount. We will multiply the number of apple slices to 1 1/2 2 2/3 X 1 1/2 8/3 X3/2 = 24/6 = 4 cups Zoe will use 4 cups of apple slices.

Question 5. Edgar has 2.8 meters of rope. If he cuts it into 7 equal parts, how long will each piece be? ______ meters

Answer: 0.4 meters

Explanation: 2.8/7 = 0.4 meters

Question 6. Kami has 7 liters of water to fill water bottles that each hold 2.8 liters. How many bottles can she fill? ______ bottles

Answer: 2 bottles

Explanation: 7/2.8 = 2.5 she can only fill 2 because anything over that would be 8.4 liters of water

Share and Show – Page No. 109

Estimate. Then find the quotient.

Question 1. \(\frac{5}{6}\) ÷ 3 \(\frac{□}{□}\)

Explanation: 5/6 = 0.8333 is closer to 0.9 0.9/3 = 0.3 = 3/10

Use a number line to find the quotient.

Question 2. \(\frac{3}{4} \div \frac{1}{8}\) _______

Explanation: 3/4 x 8 = 3 x 2 = 6

Question 3. \(\frac{3}{5} \div \frac{3}{10}\) _______

Explanation: 3/5 x 10/3 = 2

Estimate. Then write the quotient in simplest form.

Question 4. \(\frac{3}{4} \div \frac{5}{6}\) \(\frac{□}{□}\)

Answer: \(\frac{1}{1}\)

Explanation: 3/4 = 0.75 is closer to 0.8 5/6 = 0.8333 is closer to 0.8 0.8/0.8 = 1

Practice and Homework Lesson 2.7 Question 5. \(3 \div \frac{3}{4}\) _______

Explanation: 3/4 = 0.75 3/0.75 = 4

Question 6. \(\frac{1}{2} \div \frac{3}{4}\) \(\frac{□}{□}\)

Answer: \(\frac{625}{1000}\)

Explanation: 1/2 = 0.5 3/4 = 0.75 is closer to 0.8 0.5/0.8 = 0.625 = 625/1000

Question 7. \(\frac{5}{12} \div 3\) \(\frac{□}{□}\)

Answer: \(\frac{2}{10}\)

Explanation: 5/12 = 0.4166 is closer to 0.6 0.6/3 = 0.2 = 2/10

Practice: Copy and Solve Estimate. Then write the quotient in simplest form

Question 8. \(2 \div \frac{1}{8}\) _______

Explanation: 1/8 = 0.125 is closer to 0.1 2/0.1 = 20

Question 9. \(\frac{3}{4} \div \frac{3}{5}\) \(\frac{□}{□}\)

Explanation: 3/4 = 0.75 is closer to 0.8 3/5 = is 0.6 closer to 0.8 0.8/0.8 = 1

Question 10. \(\frac{2}{5} \div 5\) \(\frac{□}{□}\)

Explanation: 2/5 = 0.4 is closer to 0.5 0.5/5 = 0.1 = 1/10

Question 11. \(4 \div \frac{1}{7}\) _______

Explanation: 1/7 = 0.1428 is closer to 0.1 4/0.1 = 40

Practice: Copy and Solve Evaluate using the order of operations.

Write the answer in simplest form.

Question 12. \(\left(\frac{3}{5}+\frac{1}{10}\right) \div 2\) \(\frac{□}{□}\)

Answer: \(\frac{7}{20}\)

Explanation: 3/5 + 1/10 = 7/10 = 0.7 0.7/2 = 7/20

Question 13. \(\frac{3}{5}+\frac{1}{10} \div 2\) \(\frac{□}{□}\)

Answer: \(\frac{13}{20}\)

Explanation: \(\frac{3}{5}+\frac{1}{10} \div 2\) (1/10)/2 = 1/20 3/5 + 1/20 = 0.65 = 13/20

Question 14. \(\frac{3}{5}+2 \div \frac{1}{10}\) _______ \(\frac{□}{□}\)

Explanation: 2/(1/10) = 1/5 3/5 + 1/5 = 4/5

Question 15. Generalize Suppose the divisor and the dividend of a division problem are both fractions between 0 and 1, and the divisor is greater than the dividend. Is the quotient less than, equal to, or greater than 1? Type below: __________

Answer: Divisor and Dividend are fractions lying between 0 and 1 Also, Divisor > Dividend A smaller number is being divided by a larger number Whenever a smaller number is divided by a larger number, the quotient is less than 1 Example: 0,5/0,6 Here, they are both numbers between 0 and 1, and the divisor is greater than the dividend. The result is 0,8333, LESS THAN 1 Hence, the answer is that the quotient will be less than 1

Problem Solving + Applications – Page No. 110

Question 16. Kristen wants to cut ladder rungs from a 6 ft board. How many ladder rungs can she cut? _______ ladder rungs

Answer: 8 ladder rungs

Explanation: Kristen wants to cut ladder rungs from a 6 ft board. ladder rungs = 3/4 ft 6/(3/4) = 8 rungs

Question 17. Pose a Problem Look back at Problem 16. Write and solve a new problem by changing the length of the board Kristen is cutting for ladder rungs. Type below: __________

Answer: Kristen wants to cut ladder rungs from a 9 ft board. How many ladder rungs can she cut? Kristen wants to cut ladder rungs from a 9 ft board. ladder rungs = 3/4 ft 9/(3/4) = 12 rungs

Question 18. Dan paints a design that has 8 equal parts along the entire length of the windowsill. How long is each part of the design? \(\frac{□}{□}\) yards

Answer: \(\frac{1}{16}\) yards

Explanation: Dan paints a design that has 8 equal parts along the entire length of the windowsill. (1/2)/8 = 1/2 x 1/8 = 1/16 yards

Question 19. Dan has a board that is \(\frac{15}{16}\) yd. How many “Keep Out” signs can he make if the length of the sign is changed to half of the original length? _______ signs

Answer: 3 signs

Explanation: Dan has a board that is \(\frac{15}{16}\) yd. If the length of the sign is changed to half of the original length, (5/8)/2 = 5/16 (15/16) ÷ 5/16 = 15/16 x 16/5 = 3

Question 20. Lauren has \(\frac{3}{4}\) cup of dried fruit. She puts the dried fruit into bags, each holding \(\frac{1}{8}\) cup. How many bags will Lauren use? Explain your answer using words and numbers. Type below: __________

Explanation: Lauren has \(\frac{3}{4}\) cup of dried fruit. She puts the dried fruit into bags, each holding \(\frac{1}{8}\) cup. 3/4 ÷ 1/8 = 3/4 x 8 = 6 Lauren has 3/4 and in 1/4 there are 2 1/8s. That 3 fourths times two = 6 so 6 one eights

Divide Fractions – Page No. 111

Question 1. \(5 \div \frac{1}{6}\) _____

Explanation: 1/6 = 0.166 is closer to 0.2 5/0.2 = 25

Question 2. \(\frac{1}{2} \div \frac{1}{4}\) _____

Explanation: 1/2 = 0.5 is closer to 1 1/4 = 0.25 is closer to 0.2 1/0.2 = 5

Question 3. \(\frac{4}{5} \div \frac{2}{3}\) _____ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{5}\)

Explanation: 4/5 = 0.8 is closer to 0.8 2/3 = 0.66 is closer to 0.6 0.8/0.6 = 1 1/5

Question 4. \(\frac{14}{15} \div 7\) \(\frac{□}{□}\)

Answer: \(\frac{2}{15}\)

Explanation: 14/15 = 0.9333 0.9/7 = 2/15

Question 5. \(8 \div \frac{1}{3}\) _____

Explanation: 1/3 = 0.33 is closer to 0.4 8/0.4 = 20

Question 6. \(\frac{12}{21} \div \frac{2}{3}\) \(\frac{□}{□}\)

Explanation: 12/21 = 0.571 is closer to 0.6 2/3 = 0.666 is closer to 0.6 0.6/0.6 = 1

Question 7. \(\frac{5}{6} \div \frac{5}{12}\) _____

Explanation: 5/6 = 0.833 is closer to 0.8 5/12 = 0.416 is closer to 0.4 0.8/0.4 = 2

Question 8. \(\frac{5}{8} \div \frac{1}{2}\) _____ \(\frac{□}{□}\)

Answer: 1 \(\frac{2}{10}\)

Explanation: 5/8 = 0.625 is closer to 0.6 1/2 = 0.5 is closer to 0.5 0.6/0.5 = 1.2 = 1 2/10

Question 9. Joy ate \(\frac{1}{4}\) of a pizza. If she divides the rest of the pizza into pieces equal to \(\frac{1}{8}\) pizza for her family, how many pieces will her family get? _____ pieces

Answer: 6 pieces

Explanation: The pizza is divided into 4 pieces, Joy ate 1/4. So, the left pieces are 1 – 1/4 = 3/4 Now, 3/4 of a pizza and Joy will divide the rest of the pizza in pieces equal to 1/8, so we need to make a division (3/4) ÷ (1/8) = 24/4 = 6 pieces.

Question 10. Hideko has \(\frac{3}{5}\) yard of ribbon to tie on balloons for the festival. Each balloon will need \(\frac{3}{10}\) yard of ribbon. How many balloons can Hideko tie with ribbon? _____ balloons

Answer: 2 balloons

Explanation: 3/10 yard of ribbon required to tie = 1 balloon 3/5 yard of ribber can tie = (3/5) ÷ (3/10) = 2 ballons With 3/5 yard, Hideko can tie 2 balloons

Question 11. Rick knows that 1 cup of glue weighs \(\frac{1}{18}\) pound. He has \(\frac{2}{3}\) pound of glue. How many cups of glue does he have? _____ cups

Answer: 12 cups

Explanation: For 1/18lb, 1 cup For 2/3lb, x cups. 1/8x = 1 x 2/3 1/8x = 2/3 x = 2/3 x 18 x = 2 x 6 = 12 cups

Question 12. Mrs. Jennings had \(\frac{5}{7}\) gallon of paint. She gave \(\frac{1}{7}\) gallon each to some students. How many students received paint if Mrs. Jennings gave away all the paint? _____ students

Answer: 4 students

Explanation: Mrs. Jennings had \(\frac{5}{7}\) gallon of paint. She gave \(\frac{1}{7}\) gallon each to some students. \(\frac{5}{7}\) ÷ \(\frac{1}{7}\) = 25/7 = 3.571 is closer to 4

Question 13. Write a word problem that involves two fractions. Include the solution. Type below: __________

Answer: Mrs. Jennings had \(\frac{5}{7}\) gallon of paint. She gave \(\frac{1}{7}\) gallon each to some students. How many students received paint if Mrs. Jennings gave away all the paint? Answer: Mrs. Jennings had \(\frac{5}{7}\) gallon of paint. She gave \(\frac{1}{7}\) gallon each to some students. \(\frac{5}{7}\) ÷ \(\frac{1}{7}\) = 25/7 = 3.571 is closer to 4

Lesson Check – Page No. 112

Question 1. There was \(\frac{2}{3}\) of a pizza for 6 friends to share equally. What fraction of the pizza did each person get? \(\frac{□}{□}\)

Explanation: There was \(\frac{2}{3}\) of a pizza for 6 friends to share equally. \(\frac{2}{3}\) ÷ 6 = 2/3 x 1/6 = 2/18 = 1/9

Question 2. Rashad needs \(\frac{2}{3}\) pound of wax to make a candle. How many candles can he make with 6 pounds of wax? _____ candles

Answer: 9 candles

Explanation: Rashad needs 2/3 pound a wax to make candles. 1 Candle = 2/3 pounds. So, for 2 pounds, 3 x 2/3 = 3 candles 2 pounds = 3 candles 1 pound = 3/2 candles So, for 6 pounds, 6 x 3/2 = 9 candles

Question 3. Jeremy had \(\frac{3}{4}\) of a submarine sandwich and gave his friend \(\frac{1}{3}\) of it. What fraction of the sandwich did the friend receive? \(\frac{□}{□}\)

Explanation: Jeremy had \(\frac{3}{4}\) of a submarine sandwich and gave his friend \(\frac{1}{3}\) of it. 1/3 x 3/4 = 1/4

Question 4. Ebony walked at a rate of 3 \(\frac{1}{2}\) miles per hour for 1 \(\frac{1}{3}\) hours. How far did she walk? _____ \(\frac{□}{□}\)

Answer: 4 \(\frac{2}{3}\)

Explanation: Ebony walked at a rate of 3 \(\frac{1}{2}\) miles per hour for 1 \(\frac{1}{3}\) hours. 3 1/2 miles = 7/2 miles … 1 hour x miles = ? … 1 1/3 hours = 4/3 hours 7/2 x 4/3 = 1 x x x = 7/2 x 4/3 x = 14/3 = 4 2/3 miles The correct result would be 4 2/3 miles.

Question 5. Penny uses \(\frac{3}{4}\) yard of fabric for each pillow she makes. How many pillows can she make using 6 yards of fabric? _____ pillows

Answer: 8 pillows

Explanation: Penny uses \(\frac{3}{4}\) yard of fabric for each pillow she makes. Using 6 yards of fabric 6/(3/4) = 24/3 = 8

Question 6. During track practice, Chris ran 2.5 laps in 81 seconds. What was his average time per lap? _____ seconds

Answer: 32.4 seconds

Explanation: During track practice, Chris ran 2.5 laps in 81 seconds. 81/2.5 = 32.4 seconds

Share and Show – Page No. 115

Explanation: Model 3 with 3 hexagonal blocks. Model 1/2 with 1 trapezoid block. For 1/6, 6 triangle blocks are equal to 1 hexagon. So, a triangle block shows 1/6. Count the triangles. There are 21 triangle blocks. So, 3 1/2 ÷ 1/6 = 21.

Explanation: Model 2 with 2 hexagonal blocks. Model 1/2 with 1 trapezoid block. For 1/6, 6 triangle blocks are equal to 1 hexagon. So, a triangle block shows 1/6. Count the triangles. There are 15 triangle blocks. So, \(2 \frac{1}{2} \div \frac{1}{6}\) = 15.

Use pattern blocks to find the quotient. Then draw the model.

Question 3. \(2 \frac{2}{3} \div \frac{1}{6}\) _____

Explanation: 2 2/3 = 8/3 8/3 ÷ 1/6 = 16

Question 4. \(3 \frac{1}{2} \div \frac{1}{2}\) _____

Explanation: 3 1/2 = 7/2 7/2 ÷ 1/2 = 7

Draw a model to find the quotient.

Question 5. \(3 \frac{1}{2} \div 3\) _____ \(\frac{□}{□}\)

Explanation: 3 1/2 = 7/2 7/2 ÷ 3 = 21/2

Question 6. \(1 \frac{1}{4} \div 2\) \(\frac{□}{□}\)

Explanation: 1/4 ÷ 2 = 1/2

Question 7. Use Appropriate Tools Explain how models can be used to divide mixed numbers by fractions or whole numbers Type below: __________

Answer: Multiply the whole number part by the fraction’s denominator. Add that to the numerator. Then write the result on top of the denominator.

Problem Solving + Applications – Page No. 116

Use a model to solve. Then write an equation for the model.

Question 8. Use Models Eliza opens a box of bead kits. The box weighs 2 \(\frac{2}{3}\) lb. Each bead kit weighs \(\frac{1}{6}\) lb. How many kits are in the box? What does the answer mean? Type below: __________

Explanation: Eliza opens a box of bead kits. The box weighs 2 \(\frac{2}{3}\) lb. Each bead kit weighs \(\frac{1}{6}\) lb, 2 \(\frac{2}{3}\) ÷ \(\frac{1}{6}\) = 8/3 ÷ 1/6 = 16. 16 kits are in the box

Question 9. Hassan has two boxes of trail mix. Each box holds 1 \(\frac{2}{3}\) lb of trail mix. He eats \(\frac{1}{3}\) lb of trail mix each day. How many days can Hassan eat trail mix before he runs out? _____ days

Answer: 10 days

Explanation: Hassan has two boxes of trail mix. Each box holds 1 \(\frac{2}{3}\) lb of trail mix. 1 \(\frac{2}{3}\) = 5/3 2 x (5/3) = 10/3 He eats \(\frac{1}{3}\) lb of trail mix each day. 10/3 ÷ 1/3 = 10 Hassan eats trail mix for 10 days before he runs out.

Answer: \(2 \frac{1}{3} \div \frac{1}{6}\) = 7/3 ÷ 1/6 = 14. He said the quotient is 7. His answer is Nonsense.

Question 11. Eva is making muffins to sell at a fundraiser. She has 2 \(\frac{1}{4}\) cups of flour, and the recipe calls for \(\frac{3}{4}\) cup of flour for each batch of muffins. Explain how to use a model to find the number of batches of muffins Eva can make. Type below: __________

Explanation: Eva is making muffins to sell at a fundraiser. She has 2 \(\frac{1}{4}\) cups of flour, and the recipe calls for \(\frac{3}{4}\) cup of flour for each batch of muffins. 2 \(\frac{1}{4}\) ÷ \(\frac{3}{4}\) = 9/4 ÷ 3/4 = 3

Model Mixed Number Division – Page No. 117

Explanation: Count the number of trapezoids to find the answer.

Use pattern blocks or another model to find the quotient. Then draw the model.

Question 3. \(2 \frac{1}{2} \div \frac{1}{6}\) _____

Explanation: Model 2 with 2 hexagonal blocks. Model 1/2 with 1 trapezoid block. For 1/6, 6 triangle blocks are equal to 1 hexagon. So, a triangle block shows 1/6. Count the triangles. There are 15 triangle blocks. So, 212÷16 = 15.

Question 4. \(2 \frac{3}{4} \div 2\) _____

Explanation: 2 3/4 ÷ 2 = 11/2

Question 5. Marty has 2 \(\frac{4}{5}\) quarts of juice. He pours the same amount of juice into 2 bottles. How much does he pour into each bottle? _____ \(\frac{□}{□}\) quarts

Answer: 1\(\frac{2}{5}\) quarts

Explanation: Marty has 2 \(\frac{4}{5}\) quarts of juice. He pours the same amount of juice into 2 bottles. 2 \(\frac{4}{5}\) = 14/5 = 2.8 2.8/2 = 1.4 = 1 2/5

Question 6. How many \(\frac{1}{3}\) pound servings are in 4 \(\frac{2}{3}\) pounds of cheese? _____ pounds

Answer: 14 pounds

Explanation: 4 2/3 = 14/3 (14/3)/(1/3) = 14

Question 7. Write a word problem that involves dividing a mixed number by a whole number. Solve the problem and describe how you found the answer. Type below: __________

Answer: How many \(\frac{1}{3}\) pound servings are in 4 \(\frac{2}{3}\) pounds of cheese? Explanation: 4 2/3 = 14/3 (14/3)/(1/3) = 14

Lesson Check – Page No. 118

Sketch a model to find the quotient.

Question 1. Emma has 4 \(\frac{1}{2}\) pounds of birdseed. She wants to divide it evenly among 3 bird feeders. How much birdseed should she put in each? _____ \(\frac{□}{□}\) pounds

Answer: 1\(\frac{1}{2}\) pounds

Explanation: Emma has 4 1/2 pounds of birdseed. Convert this to an improper fraction. 4 1/2 = 9/2 Emma wants to divide it evenly among 3 bird feeders. So, she should put (9/2)/3 = 3/2 = 1 1/2

Question 2. A box of crackers weighs 11 \(\frac{1}{4}\) ounces. Kaden estimates that one serving is \(\frac{3}{4}\) ounce. How many servings are in the box? _____ servings

Answer: 15 servings

Explanation: 11 1/4 by 3/4 11 1/4 = 45/4 45/4 / 3/4 = 45/4 × 4/3 = 180/12 = 15 there are 15 servings

Question 3. The Ecology Club has volunteered to clean up 4.8 kilometers of highway. The members are organized into 16 teams. Each team will clean the same amount of highway. How much highway will each team clean? _____ kilometers

Answer: 0.3 kilometers

Explanation: The Ecology Club has volunteered to clean up 4.8 kilometers of highway. The members are organized into 16 teams. The total length of the highway is given to clean = 4.8 kilometers If the members are organized into 16 teams. 4.8/16 = 0.3 Hence, each team will clean 0.3 kilometers of the highway.

Question 4. Tyrone has $8.06. How many bagels can he buy if each bagel costs $0.65? _____ bagels

Answer: 12 bagels

Explanation: $8.06/$0.65 = 12.4 12 bagels

Question 5. A nail is 0.1875 inches thick. What is its thickness as a fraction? Is 0.1875 inch closer to \(\frac{1}{8}\) inch or \(\frac{1}{4}\) inch on a number line? Type below: __________

Answer: 0.1875 = 3/16 which is at the same distance to 1/4 and 1/8 It is the same distance apart.

Question 6. Maria wants to find the product of 5 \(\frac{3}{20}\) × 3 \(\frac{4}{25}\) using decimals instead of fractions. How can she rewrite the problem using decimals? Type below: __________

Answer: 16.274

Explanation: The decimal for 5 3/20 is 5.15 The decimal for 3 4/25 is 3.16 5.15 × 3.16 = 16.274

Share and Show – Page No. 121

Question 1. \(4 \frac{1}{3} \div \frac{3}{4}\) ______ \(\frac{□}{□}\)

Answer: 5\(\frac{375}{1000}\)

Explanation: 4 1/3 = 13/3 = 4.333 is closer to 4.3 3/4 = 0.75 is closer to 0.8 4.3/0.8 = 5.375 = 5 375/1000

Question 2. Six hikers shared 4 \(\frac{1}{2}\) lb of trail mix. How much trail mix did each hiker receive? \(\frac{□}{□}\)

Answer: \(\frac{75}{100}\)

Explanation: 6 hikers = 4.5 lbs of trail mix 4.5/6= .75 lbs each hiker.

Question 3. \(5 \frac{2}{3} \div 3\) ______ \(\frac{□}{□}\)

Answer: 2\(\frac{947}{1000}\)

Explanation: 5 2/3 = 17/3 = 5.666 is closer to 5.6 5.6/3 = 1.866 is closer to 1.9 5.6/1.9 = 2.947 = 2 947/1000

Question 4. \(7 \frac{1}{2} \div 2 \frac{1}{2}\) ______

Explanation: 7 1/2 = 15/2 = 7.5 2 1/2 = 5/2 = 2.5 7.5/2.5 = 3

Question 5. \(5 \frac{3}{4} \div 4 \frac{1}{2}\) ______ \(\frac{□}{□}\)

Answer: 1\(\frac{27}{100}\)

Explanation: 5 3/4 = 23/4 = 5.75 4 1/2 = 9/2 = 4.5 5.75/4.5 = 1.27 = 1 27/100

Question 6. \(5 \div 1 \frac{1}{3}\) ______ \(\frac{□}{□}\)

Answer: 3\(\frac{84}{100}\)

Explanation: 1 1/3 = 4/3 = 1.33 is closer to 1.3 5/1.3 = 3.84 = 3 84/100

Divide Mixed Numbers Lesson 2.9 Question 7. \(6 \frac{3}{4} \div 2\) ______ \(\frac{□}{□}\)

Answer: 3\(\frac{2}{5}\)

Explanation: 6 3/4 = 27/4 = 6.75 is closer to 6.8 6.8/2 = 3.4 = 3 2/5

Question 8. \(2 \frac{2}{9} \div 1 \frac{3}{7}\) ______ \(\frac{□}{□}\)

Answer: 1\(\frac{571}{1000}\)

Explanation: 2 2/9 = 20/9 = 2.22 is closer to 2.2 1 3/7 = 10/7 = 1.428 is closer to 1.4 2.2/1.4 = 1.571 = 1 571/1000

Question 9. How many 3 \(\frac{1}{3}\) yd pieces can Amanda get from a 3 \(\frac{1}{3}\) yd ribbon? ______

Explanation: (3 1/3) ÷ (3 1/3) = 1

Question 10. Samantha cut 6 \(\frac{3}{4}\) yd of yarn into 3 equal pieces. Explain how she could use mental math to find the length of each piece Type below: __________

Answer: 27/12

Explanation: Samantha cut 6 \(\frac{3}{4}\) yd of yarn into 3 equal pieces. 6 3/4 = 27/4 (27/4)/3 (27/4)(1/3) = 27/12

Evaluate Algebra Evaluate using the order of operations. Write the answer in simplest form.

Question 11. \(1 \frac{1}{2} \times 2 \div 1 \frac{1}{3}\) _____ \(\frac{□}{□}\)

Answer: 2\(\frac{1}{4}\)

Explanation: (1 1/2) × 2 = 3/2 × 2 = 3 1 1/3 = 4/3 3/(4/3) = 9/4 = 2.25 = 2 1/4

Question 12. \(1 \frac{2}{5} \div 1 \frac{13}{15}+\frac{5}{8}\) _____ \(\frac{□}{□}\)

Answer: 1\(\frac{3}{8}\)

Explanation: (1 2/5)/(1 13/15) = (7/5)/(28/15) = 3/4 = 0.75 0.75 + 0.625 = 1.375 = 1 3/8

Question 13. \(3 \frac{1}{2}-1 \frac{5}{6} \div 1 \frac{2}{9}\) _____

Explanation: (1 5/6)/(1 2/9) = (11/6)/11/9 = 3/2 = 1 1/2 = 1.5 3 1/2 = 7/2 = 3.5 3.5 – 1.5 = 2

Question 14. Look for a Pattern Find these quotients: \(20 \div 4 \frac{4}{5}\), \(10 \div 4 \frac{4}{5}\), \(5 \div 4 \frac{4}{5}\). Describe a pattern you see. Type below: __________

Answer: 20 ÷ 4 4/5 = 20 ÷ 24/5 = 20/4.8 = 4.1666 10 ÷ 4 4/5 = 10 ÷ 24/5 = 10/4.8 = 2.08333 5 ÷ 4 4/5 = 5 ÷ 24/5 = 5/4.8 = 1.04166 The pattern is multiplied by 2 every time.

Page No. 122

Answer: How many breaks Dina will take when hikes \(\frac{1}{2}\) of the easy trail and stops for a break every 3 \(\frac{1}{4}\) mile.

Question 15. b. How will you use the information in the table to solve the problem? Type below: __________

Answer: Dina easy trail length, break time

Question 15. c. How can you find the distance Dina hikes? How far does she hike? ______ \(\frac{□}{□}\) miles

Answer: 9\(\frac{3}{4}\) miles

Explanation: 19 1/2 × 1/2 = 39/2 × 1/2 = 39/4 = 9 3/4

Question 15. d. What operation will you use to find how many breaks Dina takes? Type below: __________

Answer: Division

Question 15. e. How many breaks will Dina take? ______ breaks

Answer: 3 breaks

Explanation: 39/4 ÷ 13/4 = 3

Question 16. Carlo packs 15 \(\frac{3}{4}\) lb of books in 2 boxes. Each book weighs 1 \(\frac{1}{8}\) lb. There are 4 more books in Box A than in Box B. How many books are in Box A? Explain your work. ______ books

Answer: Carlo packs 15 \(\frac{3}{4}\) lb of books in 2 boxes. Each book weighs 1 \(\frac{1}{8}\) lb. 15 \(\frac{3}{4}\) ÷ 1 \(\frac{1}{8}\) = 63/4 ÷ 9/8 = 14 14 books available in 2 boxes. There are 4 more books in Box A than in Box B. Box A contains 5 + 4 = 9 books Box B contains 5 books

Question 17. Rex’s goal is to run 13 \(\frac{3}{4}\) miles over 5 days. He wants to run the same distance each day. Jordan said that Rex would have to run 3 \(\frac{3}{4}\) miles each day to reach his goal. Do you agree with Jordan? Explain your answer using words and numbers. Type below: __________

Answer: Rex’s goal is to run 13 \(\frac{3}{4}\) miles over 5 days. He wants to run the same distance each day. 13 \(\frac{3}{4}\) ÷ 5 = 55/4 ÷ 5 = 11/4 or 2 3/4. Jordan answer is wrong

Divide Mixed Numbers – Page No. 123

Question 1. \(2 \frac{1}{2} \div 2 \frac{1}{3}\) ______ \(\frac{□}{□}\)

Answer: 1\(\frac{1}{2}\)

Explanation: 2 1/2 = 5/2 = 2.5 is closer to 3 2 1/3 = 7/3 = 2.333 is closer to 2 3/2 = 1.5 = 1 1/2

Question 2. \(2 \frac{2}{3} \div 1 \frac{1}{3}\) ______

Explanation: 2 2/3 = 8/3 = 2.666 is closer to 2.6 1 1/3 = 4/3 = 1.333 is closer to 1.3 2.6/1.3 = 2

Question 3. \(2 \div 3 \frac{5}{8}\) \(\frac{□}{□}\)

Explanation: 3 5/8 = 29/8 = 3.625 is closer to 3.6 2/3.6 = 0.5 = 1/2

Question 4. \(1 \frac{13}{15} \div 1 \frac{2}{5}\) \(\frac{□}{□}\)

Answer: \(\frac{126}{100}\)

Explanation: 1 13/15 = 28/15 = 1.8666 is closer to 1.9 1 2/5 = 7/5 = 1.4 is closer to 1.5 1.9/1.5 = 1.266 126/100

Question 5. \(10 \div 6 \frac{2}{3}\) ______ \(\frac{□}{□}\)

Explanation: 6 2/3 = 20/3 = 6.666 is closer to 6.7 10/6.7 = 3/2 = 1 1/2

Question 6. \(2 \frac{3}{5} \div 1 \frac{1}{25}\) ______ \(\frac{□}{□}\)

Answer: 2\(\frac{3}{5}\)

Explanation: 2 3/5 = 13/5 = 2.6 1 1/25 = 26/25 = 1.04 is closer to 1 2.6/1 = 13/5 or 2 3/5

Question 7. \(2 \frac{1}{5} \div 2\) ______ \(\frac{□}{□}\)

Answer: 1\(\frac{1}{10}\)

Explanation: 2 1/5 = 11/5 = 2.2 is closer to 2.2 2.2/2 = 1.1 = 11/10 = 1 1/10

Lesson 2.9 Divide Mixed Numbers Question 8. Sid and Jill hiked 4 \(\frac{1}{8}\) miles in the morning and 1 \(\frac{7}{8}\) miles in the afternoon. How many times as far did they hike in the morning as in the afternoon? ______ \(\frac{□}{□}\) times

Answer: 2\(\frac{1}{5}\) times

Explanation: Sid and Jill hiked 4 \(\frac{1}{8}\) miles in the morning and 1 \(\frac{7}{8}\) miles in the afternoon. 4 \(\frac{1}{8}\) = 33/8 1 \(\frac{7}{8}\) = 15/8 (33/8) ÷ (15/8) = 33/15 = 11/5 or 2 1/5

Question 9. It takes Nim 2 \(\frac{2}{3}\) hours to weave a basket. He worked Monday through Friday, 8 hours a day. How many baskets did he make? ______ baskets

Answer: 15 baskets

Explanation: he worked (Mon – Fri) 5 days at 8 hrs per day = 5 × 8= 40 hrs 40/ (2 2/3) = 40 / (8/3) = 40 × 3/8 = 120/8 = 15 baskets

Question 10. A tree grows 1 \(\frac{3}{4}\) feet per year. How long will it take the tree to grow from a height of 21 \(\frac{1}{4}\) feet to a height of 37 feet? ______ years

Answer: 9 years

Explanation: A tree grows 1 3/4 = 7/4 feet per year. If you would like to know how long will it take the tree to grow from a height of 21 1/4 = 85/4 feet to a height of 37 feet, 37 – 21 1/4 = 37 – 85/4 = 148/4 – 85/4 = 63/4 = 15 3/4 15 3/4 / 1 3/4 = 63/4 / 7/4 = 63/4 × 4/7 = 9 years

Question 11. Explain how you would find how many 1 \(\frac{1}{2}\) cup servings there are in a pot that contains 22 \(\frac{1}{2}\) cups of soup. Type below: __________

Answer: Given that, Total number of cups = 22 1/2 The number of cups required for each serving = 1 1/2 The number of servings = 22 1/2 ÷ 1 1/2 = 45/2 ÷ 3/2 = 45/3 = 15

Lesson Check – Page No. 124

Question 1. Tom has a can of paint that covers 37 \(\frac{1}{2}\) square meters. Each board on the fence has an area of \(\frac{3}{16}\) square meters. How many boards can he paint? ______ boards

Answer: 200 boards

Explanation: Tom has a can of paint that covers 37 \(\frac{1}{2}\) square meters. Each board on the fence has an area of \(\frac{3}{16}\) square meters. 37 \(\frac{1}{2}\) ÷ \(\frac{3}{16}\) = 200 square meters

Question 2. A baker wants to put 3 \(\frac{3}{4}\) pounds of apples in each pie she makes. She purchased 52 \(\frac{1}{2}\) pounds of apples. How many pies can she make? ______ pies

Answer: 14 pies

Explanation: A baker wants to put 3 \(\frac{3}{4}\) pounds of apples in each pie she makes. She purchased 52 \(\frac{1}{2}\) pounds of apples. 52 \(\frac{1}{2}\) ÷ 3 \(\frac{3}{4}\) = 14 pies

Question 3. The three sides of a triangle measure 9.97 meters, 10.1 meters, and 0.53 meters. What is the distance around the triangle? ______ meters

Answer: 20.6 meters

Explanation: The distance around the triangle is called the perimeter, to get it we must add the 3 sides. So, 9.97 + 10.1 + 0.53 = 20.6 meters

Question 4. Selena bought 3.75 pounds of meat for $4.64 per pound. What was the total cost of the meat? $ ______

Answer: $17.40

Explanation: Selena bought 3.75 pounds of meat. The cost of meat of one pound = $4.64 The total cost of the meat = 4.64 × 3.75 = $17.40 The total cost of 3.75 lb of meat was $17.40.

Question 5. Melanie prepared 7 \(\frac{1}{2}\) tablespoons of a spice mixture. She uses \(\frac{1}{4}\) tablespoon to make a batch of barbecue sauce. Estimate the number of batches of barbecue sauce she can make using the spice mixture. Type below: __________

Answer: 30 batches of sauce

Explanation: Melanie prepared 7 \(\frac{1}{2}\) tablespoons of a spice mixture. She uses \(\frac{1}{4}\) tablespoon to make a batch of barbecue sauce. 4 X 1/4 tbsp = 1 tbsp. 4 X 7 1/2 = 30. She can make 30 batches of sauce

Question 6. Arturo mixed together 1.24 pounds of pretzels, 0.78 pounds of nuts, 0.3 pounds of candy, and 2 pounds of popcorn. He then packaged it in bags that each contained 0.27 pounds. How many bags could he fill? ______ bags

Answer: 16 bags

Explanation: Arturo mixed together 1.24 pounds of pretzels, 0.78 pounds of nuts, 0.3 pounds of candy, and 2 pounds of popcorn. 1.24 + 0.78 + 0.3 + 2 = 4.32 4.32/0.27 = 16

Page No. 127

Question 1. There is \(\frac{4}{5}\) lb of sand in the class science supplies. If one scoop of sand weighs \(\frac{1}{20}\) lb, how many scoops of sand can Maria get from the class supplies and still leave \(\frac{1}{2}\) lb in the supplies? Type below: __________

Answer: 16 scoops

Explanation: There is \(\frac{4}{5}\) lb of sand in the class science supplies. If one scoop of sand weighs \(\frac{1}{20}\) lb, \(\frac{4}{5}\) ÷ \(\frac{1}{20}\) = 4/5 × 1/20 = 16 scoops

Question 2. What if Maria leaves \(\frac{2}{5}\) lb of sand in the supplies? How many scoops of sand can she get? ______ scoops

Answer: 8 scoops

Explanation: There is \(\frac{2}{5}\) lb of sand in the class science supplies. If one scoop of sand weighs \(\frac{1}{20}\) lb, \(\frac{2}{5}\) ÷ \(\frac{1}{20}\) = 2/5 × 20 = 8

Question 3. There are 6 gallons of distilled water in the science supplies. If 10 students each use an equal amount of distilled water and there is 1 gal left in the supplies, how much will each student get? \(\frac{□}{□}\) gallon

Answer: \(\frac{1}{2}\) gallon

Explanation: There are 6 gallons of distilled water in the science supplies. There is 1 gal left in the supplies, 6 – 1 = 5 10 students each use an equal amount of the distilled water = 5/10 = 1/2 .5 gal for each student

On Your Own – Page No. 128

Question 4. The total weight of the fish in a tank of tropical fish at Fish ‘n’ Fur was \(\frac{7}{8}\) lb. Each fish weighed \(\frac{1}{64}\) lb. After Eric bought some fish, the total weight of the fish remaining in the tank was \(\frac{1}{2}\) lb. How many fish did Eric buy? ______ fish

Answer: 386 fish

Explanation: The total weight of the fish in a tank of tropical fish at Fish ‘n’ Fur was \(\frac{7}{8}\) lb. Each fish weighed \(\frac{1}{64}\) lb. After Eric bought some fish, the total weight of the fish remaining in the tank was \(\frac{1}{2}\) lb. 386 is the answer

Question 5. Fish ‘n’ Fur had a bin containing 2 \(\frac{1}{2}\) lb of gerbil food. After selling bags of gerbil food that each held \(\frac{3}{4}\) lb, \(\frac{1}{4}\) lb of food was left in the bin. If each bag of gerbil food sold for $3.25, how much did the store earn? $ ______

Answer: $9.75

Explanation: The store would earn 9.75$ because 3 bags of gerbil food is sold. Then you would multiply 3 by 3.25.

Question 6. Describe Niko bought 2 lb of dog treats. He gave his dog \(\frac{3}{5}\) lb of treats one week and \(\frac{7}{10}\) lb of treats the next week. Describe how Niko can find how much is left. Type below: __________

Answer: Niko bought 2 lb of dog treats. He gave his dog \(\frac{3}{5}\) lb of treats one week and \(\frac{7}{10}\) lb of treats the next week. Let us find the amount of dog food eaten by dogs in two months. 3/5 + 7/10 = 13/10 Now we will subtract the amount of food eaten by the dog from the amount of food initially to find the remaining amount of dog food. 2 – 13/10 = 7/10 Therefore, 7/10 pounds of food was remaining in the bag at the end of the two months.

Question 7. There were 14 \(\frac{1}{4}\) cups of apple juice in a container. Each day, Elise drank 1 \(\frac{1}{2}\) cups of apple juice. Today, there is \(\frac{3}{4}\) cup of apple juice left. Derek said that Elise drank apple juice on nine days. Do you agree with Derek? Use words and numbers to explain your answer. Type below: __________

Answer: Derek is correct.

Explanation: An apple juice the container had 14 1/2 =14.25 She drank per day 1 1/2= 1.5 The left part in the container 3/4= .75 14.25 cups – .75 cup = 13.5 cups 13.5 cups ÷ 1.5 cups per day= 9 days

Problem Solving Fraction Operations – Page No. 129

Read each problem and solve.

Answer: 9 friends

Explanation: Let us say that there are x friends. Each one gets 1/18 of the original pizza: but this in turn leaves 1/6 of the 2/3 leftover. 1x/18 = 2/3 – 1/6 x = 12 – 3 = 9

Question 2. Sarah’s craft project uses pieces of yarn that are \(\frac{1}{8}\) yard long. She has a piece of yarn that is 3 yards long. How many \(\frac{1}{8}\) -yard pieces can she cut and still have 1 \(\frac{1}{4}\) yards left? ______ pieces

Answer: 14 pieces

Explanation: Sarah’s craft project uses pieces of yarn that are \(\frac{1}{8}\) yard long. She has a piece of yarn that is 3 yards long. If she left 1 \(\frac{1}{4}\) yards left, 3 – 1 \(\frac{1}{4}\) = 7/4 7/4 ÷ \(\frac{1}{8}\) = 14

Question 3. Alex opens a 1-pint container of orange butter. He spreads \(\frac{1}{16}\) of the butter on his bread. Then he divides the rest of the butter into \(\frac{3}{4}\) -pint containers. How many \(\frac{3}{4}\) -pint containers is he able to fill? ______ \(\frac{□}{□}\) containers

Answer: 1\(\frac{1}{4}\) containers

Explanation: Alex opens a 1-pint container of orange butter. He spreads \(\frac{1}{16}\) of the butter on his bread. 1 – 1/16 = 15/16 Then he divides the rest of the butter into \(\frac{3}{4}\) -pint containers. (15/16) ÷ (3/4) = 5/4 = 1 1/4

Question 4. Kaitlin buys \(\frac{9}{10}\) a pound of orange slices. She eats \(\frac{1}{3}\) of them and divides the rest equally into 3 bags. How much is in each bag? ______ lb

Answer: 17/90 lb

Explanation: Kaitlin buys \(\frac{9}{10}\) a pound of orange slices. She eats \(\frac{1}{3}\) of them and divides the rest equally into 3 bags. If she starts with 9/10 pounds and has eaten 1/3 of them, 9/10 – 1/3 = 17/30 This is the amount she has left. Let’s divide this value by 3 to see how many pounds are in one bag. (17/30)/3 = 17/90 There are 17/90 pounds in one bag.

Question 5. Explain how to draw a model that represents \(\left(1 \frac{1}{4}-\frac{1}{2}\right) \div \frac{1}{8}\). Type below: __________

Answer: Divide 2 bars into 8 quarters. Below that draw 1 1/4 or 5 quarters. Remove 1/2 or 2 quarters Divide each of the 3 quarters left into 2 eighths

Explanation: \(\left(1 \frac{1}{4}-\frac{1}{2}\right) \div \frac{1}{8}\) 1 1/4 -1/2 = 5/4 – 1/2 = 3/4 3/4 ÷ 1/8 = 6

Lesson Check – Page No. 130

Question 1. Eva wanted to fill bags with \(\frac{3}{4}\) pounds of trail mix. She started with 11 \(\frac{3}{8}\) pounds but ate \(\frac{1}{8}\) pound before she started filling the bags. How many bags could she fill? ______ bags

Answer: 15 bags

Explanation: 11 and 3/8-1/8=11 and 2/8=11 and 1/4 3/4 times x bags=11 and 1/4 convert 11 and 1/4 to improper fraction 11 and 1/4 = 11 + 1/4 = 44/4 + 1/4 = 45/4 3/4 times x bags=45/4 x bags = 45/4 × 4/3 = 15 bags she could fill 15 bags

Question 2. John has a roll containing 24 \(\frac{2}{3}\) feet of wrapping paper. He wants to divide it into 11 pieces. First, though, he must cut off \(\frac{5}{6}\) foot because it was torn. How long will each piece be? ______ \(\frac{□}{□}\) feet

Answer: 2\(\frac{4}{25}\) feet

Explanation: John had a roll containing wrapping paper = 24 2/3 = 74/3 First, he must cut off 5/6 feet because it was torn. He wants to divide it into 11 pieces. 74/3 – 5/6 Taking the L.C.M of 3 and 6 is 6 (148-5)/6 = 143/6 = 23.83 feet He wants to divide it into 11 pieces. length of the each piece = 23.83/11 = 2.16 feet

Question 3. Alexis has 32 \(\frac{2}{5}\) ounces of beads. How many necklaces can she make if each uses 2 \(\frac{7}{10}\) ounces of beads? ______ necklaces

Answer: 12 necklaces

Explanation: Alexis has 32 \(\frac{2}{5}\) ounces of beads. If each uses 2 \(\frac{7}{10}\) ounces of beads, 32 \(\frac{2}{5}\) × 2 \(\frac{7}{10}\) 32 \(\frac{2}{5}\) = 162/5 2 \(\frac{7}{10}\) = 27/10 162/5 × 27/10 = 12 necklaces

Question 4. Joseph has $32.40. He wants to buy several comic books that each cost $2.70. How many comic books can he buy? ______ comic books

Answer: 12 comic books

Explanation: Joseph has $32.40. He wants to buy several comic books that each cost $2.70. $32.40/$2.70 = 12 comic books

Question 5. A rectangle is 2 \(\frac{4}{5}\) meters wide and 3 \(\frac{1}{2}\) meters long. What is its area? ______ \(\frac{□}{□}\) m 2

Answer: 9\(\frac{4}{5}\) m2

Explanation: 2 \(\frac{4}{5}\) = 14/5 3 \(\frac{1}{2}\) = 7/2 14/5 × 7/2 = 9 4/5

Question 6. A rectangle is 2.8 meters wide and 3.5 meters long. What is its area? ______ m 2

Answer: 9.8 m 2

Explanation: A rectangle is 2.8 meters wide and 3.5 meters long. 2.8 × 3.5 = 9.8

Chapter 2 Review/Test – Page No. 131

Answer: 0.45, 0.5, 5/8, 3/4

Explanation: 3/4 = 0.75 5/8 = 0.625 0.45, 0.5 0.45 < 0.5 < 0.625 < 0.75

Question 2. For numbers 2a–2d, compare. Choose <, >, or =. 2a. 0.75 _____ \(\frac{3}{4}\) 2b. \(\frac{4}{5}\) _____ 0.325 2c. 1 \(\frac{3}{5}\) _____ 1.9 2d. 7.4 _____ 7 \(\frac{2}{5}\)

Answer: 2a. 0.75 = \(\frac{3}{4}\) 2b. \(\frac{4}{5}\) > 0.325 2c. 1 \(\frac{3}{5}\) < 1.9 2d. 7.4 = 7 \(\frac{2}{5}\)

Explanation: 2a. 3/4 = 0.75 0.75 = 0.75 2b. \(\frac{4}{5}\) = 0.8 0.8 > 0.325 2c. 1 \(\frac{3}{5}\) = 8/5 = 1.6 1.6 < 1.9 2d. 7 \(\frac{2}{5}\) = 37/5 = 7.4 7.4 = 7.4

Answer: 3a. The oak tree is the shortest. True 3b. The birch tree is the tallest. False 3c. Two of the trees are the same height. False 3d. The sycamore tree is taller than the maple tree. False

Explanation: Sycamore = 15 2/3 = 47/3 = 15.666 Oak = 14 3/4 = 59/4 = 14.75 Maple = 15 3/4 = 63/4 = 15.75 Birch = 15.72

Page No. 132

Answer: 4a. Point A represents 1.0. Yes 4b. Point B represents \(\frac{3}{10}\). Yes 4c. Point C represents 6.5. No 4d. Point D represents \(\frac{4}{5}\). Yes

Question 5. Select the values that are equivalent to one twenty-fifth. Mark all that apply. Options: a. 125 b. 25 c. 0.04 d. 0.025

Answer: c. 0.04

Explanation: one twenty-fifth = 1/25 = 0.04

Answer: a. Simplified Expression: 1/10 Product: 0.1 b. Simplified Expression: 1/2 Product: 0.5 c. Simplified Expression: 15/56 Product: 0.267 d. Simplified Expression: 1/12 Product: 0.083

Explanation: a. 2/5 × 1/4 = 2/20 Simplify using the GCF. The GCF of 2 and 20 is 2. Divide the numerator and the denominator by 2. So, 1/10 is the answer. Product: 0.1 b. 4/5 × 5/8 = 1/2 Product: 0.5 c. 3/7 × 5/8 = 15/ 56 Product: 0.267 d. 4/9 × 3/16 = 1/12 Product: 0.083

Page No. 133

Question 7. Two-fifths of the fish in Gary’s fish tank are guppies. One-fourth of the guppies are red. What fraction of the fish in Gary’s tank are red guppies? What fraction of the fish in Gary’s tank are not red guppies? Show your work. Type below: ___________

Answer: 1/10 of the fish are red guppies. and 9/10 of the fish are not red guppies.

Explanation: two-fifths of the fish in Gary’s fish tank are guppies. One-fourth of the guppies are red. Let the total number of fish in Gary’s fish tank be x. It is given that two-fifths of the fish in Gary’s fish tank are guppies. So, the number of guppies in Gary’s fish tank is 2/5 × x Given that One-fourth of the guppies are red. number of red guppies = 1/4 × 2x/5 = x/10 So, 1/10 of the fish are red guppies. 1 – 1/10 = 9/10 of the fish are not red guppies.

Question 8. One-third of the students at Finley High School play sports. Two-fifths of the students who play sports are girls. What fraction of all students are girls who play sports? Use numbers and words to explain your answer. Type below: ___________

Answer: One-third of the students at Finley High School play sports. Two-fifths of the students who play sports are girls. 1/3 × 2/5 = 2/15 of the girls in the school play sports.

Question 9. Draw a model to find the quotient. \(\frac{3}{4}\) ÷ 2 = \(\frac{3}{4}\) ÷ \(\frac{3}{8}\) = How are your models alike? How are they different? Type below: ___________

Explanation: \(\frac{3}{4}\) ÷ 2 = 3/4 × 1/2 = 3/8 \(\frac{3}{4}\) ÷ \(\frac{3}{8}\) = 3/4 × 8/3 = 2 Both models are multiplying with the 3/4. The number line model shows how many groups of 3/8 are in 3/4.

Question 10. Explain how to use a model to find the quotient. 2 \(\frac{1}{2}\) ÷ 2 = Type below: ___________

Answer: 5/4

Explanation: 2 1/2 = 5/2 5/2 groups of 2 5/2 ÷ 2 = 5/2 × 1/2 = 5/4

Page No. 134

Divide. Show your work.

Question 11. \(\frac{7}{8}\) ÷ \(\frac{3}{5}\) = _______ \(\frac{□}{□}\)

Answer: 1 \(\frac{11}{24}\)

Explanation: \(\frac{7}{8}\) ÷ \(\frac{3}{5}\) \(\frac{7}{8}\) × \(\frac{5}{3}\) = 35/24 = 1 \(\frac{11}{24}\)

Question 12. \(2 \frac{1}{10} \div 1 \frac{1}{5}=\) = _______ \(\frac{□}{□}\)

Answer: 1 \(\frac{3}{4}\)

Explanation: 2 \(\frac{1}{10}\) = 21/10 1 \(\frac{1}{5}\) = 6/5 (21/10) ÷ (6/5) = 7/4 or 1 3/4

Question 13. Sophie has \(\frac{3}{4}\) quart of lemonade. If she divides the lemonade into glasses that hold \(\frac{1}{16}\) quart, how many glasses can Sophie fill? Show your work _______ glasses

Answer: 12 glasses

Explanation: Let x be the number of glasses 1/16x = 3/4 x = 3/4 × 16 = 3 × 4 = 12 glasses

Question 14. Ink cartridges weigh \(\frac{1}{8}\) pound. The total weight of the cartridges in a box is 4 \(\frac{1}{2}\) pounds. How many cartridges does the box contain? Show your work and explain why you chose the operation you did. _______ cartridges

Answer: 36 cartridges

Explanation: Weight of ink cartridges = 1/8 pounds Total weight of the cartridges in a box = 4 1/2 = 9/2 pounds So, the number of cartridges that box contains is given by 9/2 ÷ 1/8 = 36 Hence, there are 36 cartridges that the box contains.

Question 15. Beth had 1 yard of ribbon. She used \(\frac{1}{3}\) yard for a project. She wants to divide the rest of the ribbon into pieces \(\frac{1}{6}\) yard long. How many \(\frac{1}{6}\) yard pieces of ribbon can she make? Explain your solution. _______ pieces

Answer: 4 pieces

Explanation: Beth had 1 yard of ribbon. She used \(\frac{1}{3}\) yard for a project. 1 – \(\frac{1}{3}\) = 2/3 yard left She wants to divide the rest of the ribbon into pieces \(\frac{1}{6}\) yard long. 2/3 ÷ 1/6 = 4

Page No. 135

Answer: 1/5 ÷ 3/4 = 4/15; 1/5 × 4/3 = 4/15 2/13 ÷ 1/5 = 10/13; 2/13 × 5/1 = 10/13 4/5 ÷ 3/5 = 4/3; 4/5 × 5/3 = 4/3 the product of the each pair of division and multiplication problems is the same. They are different from the operation performed.

Question 16. Part B Explain how to use the pattern in the table to rewrite a division problem involving fractions as a multiplication problem. Type below: ___________

Answer: First, since it’s the division you have to change the second fraction which is called the reciprocal. That means the second fraction has to be flipped before you can multiply the fractions.

Page No. 136

Answer: Margie hiked a 17 7/8 mile trail. Distance hiked by Margie = 17 7/8 = 143/8 mile. She stopped every 3 2/5 miles to take a picture = 17/5 mile Number of pictures = (143/8) ÷ (17/5) = 715/136 = 5.28 So she can take a maximum of 6 pictures and a minimum of 5 pictures. B is the correct answer.

Question 18. Brad and Wes are building a tree house. They cut a 12 \(\frac{1}{2}\) foot piece of wood into 5 of the same length pieces. How long is each piece of wood? Show your work. _______ \(\frac{□}{□}\) foot

Answer: 2 \(\frac{1}{2}\) foot

Explanation: Brad and Wes cut a 12 1/2 foot piece of wood into 5 of the same length. Let the length of 1 piece be x So, Length of 5 pieces = 5x The total length of wood = 25/2 5x = 25/2 x = 5/2 = 2 1/2

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Mr. Math Blog

Convert Units of Length - Lesson 6.1

Convert Units of Capacity - Lesson 6.2

Convert Units of Weight and Mass - Lesson 6.3

Transform Units - Lesson 6.4

Problem Solving - Distance, Rate, and Time - Lesson 6.5

Fractions and Decimals - Lesson 2.1

Compare and order Fractions and Decimals - Lesson 2.2

Multiply Fractions - Lesson 2.3

Simplify Factors - Lesson 2.4

Model Fraction Division - Lesson 2.5

Estimate Quotients - Lesson 2.6

Dividing Fractions - Lesson 2.7

Model Mixed Number Division - Lesson 2.8

Divide Mixed Numbers - Lesson 2.9

Problem Solving - Fraction Operations - Lesson 2.10

Understanding Positive and Negative Integers - Lesson 3.1

Compare and Order Integers - Lesson 3.2

Rational Numbers and the Number Line - Lesson 3.3

Compare and Order Rational Numbers - Lesson 3.4

Absolute Value - Lesson 3.5

Compare Absolute Value - Lesson 3.6

Rational Numbers and the Coordinate Plane - Lesson 3.7

Ordered Pair Relationships - Lesson 3.8

Distance on the Coordinate Plane - Lesson 3.9

Problem Solving - The Coordinate Plane - Lesson 3.10

Solutions of Equations - Lesson 8.1

Writing Equations - Lesson 8.2

Model and Solve Addition Equations - Lesson 8.3

Solve Addition & Subtraction Equations - Lesson 8.4

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Ratios and Rates - Lesson 4.2

Equivalent Ratios and the Multiplication Table - Lesson 4.3

Use Tables to Compare Ratios - Lesson 4.4

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Find Unit Rates - Lesson 4.6

Use Unit Rates - Lesson 4.7

Equivalent Ratios and Graphs - Lesson 4.8

Three Dimensional Figures and Nets - Lesson 11.1

Surface Area of Prisms - Lesson 11.3

Surface Area of Pyramids - Lesson 11.4

Volume of a Rectangular Prism - Lesson 11.6

Area of Parallelograms - Lesson 10.1

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Area of Trapezoids - Lesson 10.5

Area of Regular Polygons - Lesson 10.6

​Composite Figures - Lesson 10.7

Divide Multi-Digit Numbers - Lesson 1.1

Prime Factorization - Lesson 1.2

Least Common Multiple (LCM) - Lesson 1.3

Greatest Common Factor (GCF) - Lesson 1.4

Problem Solving: Apply the GCF - Lesson 1.5

Add and Subtract Decimals - Lesson 1.6

Multiply Decimals - Lesson 1.7

Divide Decimals by Whole Numbers - Lesson 1.8

Divide with Decimals - Lesson 1.9

Chapter 1 Review for Test

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Independent and Dependent Variables - Lesson 9.1

Equations and Tables - Lesson 9.2

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Find the Whole From a Percent - Lesson 5.6

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8.6: Solve Equations with Fraction or Decimal Coefficients

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Learning Objectives

Solve equations with fraction coefficients.

• Solve equations with decimal coefficients

be prepared!

Before you get started, take this readiness quiz.

• Multiply: 8 • \(\dfrac{3}{8}\). If you missed this problem, review Example 4.3.10 .
• Find the LCD of \(\dfrac{5}{6}\) and \(\dfrac{1}{4}\). If you missed this problem, review Example 4.8.1 .
• Multiply: 4.78 by 100. If you missed this problem, review Example 5.3.8 .

Solve Equations with Fraction Coefficients

Let’s use the General Strategy for Solving Linear Equations introduced earlier to solve the equation \(\dfrac{1}{8}x + \dfrac{1}{2} = \dfrac{1}{4}\).

This method worked fine, but many students don’t feel very confident when they see all those fractions. So we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.

We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but with no fractions. This process is called clearing the equation of fractions . Let’s solve the same equation again, but this time use the method that clears the fractions.

Example \(\PageIndex{1}\):

Solve: \(\dfrac{1}{8} x + \dfrac{1}{2} = \dfrac{1}{4}\).

Exercise \(\PageIndex{1}\):

Solve: \(\dfrac{1}{4} x + \dfrac{1}{2} = \dfrac{5}{8}\).

\(x = \frac{1}{2}\)

Exercise \(\PageIndex{2}\):

Solve: \(\dfrac{1}{6} y - \dfrac{1}{3} = \dfrac{1}{6}\).

Notice in Example 8.37 that once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.

HOW TO: SOLVE EQUATIONS WITH FRACTION COEFFICIENTS BY CLEARING THE FRACTIONS

Step 1. Find the least common denominator of all the fractions in the equation.

Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.

Step 3. Solve using the General Strategy for Solving Linear Equations.

Example \(\PageIndex{2}\):

Solve: 7 = \(\dfrac{1}{2} x + \dfrac{3}{4} x − \dfrac{2}{3} x\).

We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.

Exercise \(\PageIndex{3}\):

Solve: 6 = \(\dfrac{1}{2} v + \dfrac{2}{5} v − \dfrac{3}{4} v\).

Exercise \(\PageIndex{4}\):

Solve: -1 = \(\dfrac{1}{2} u + \dfrac{1}{4} u − \dfrac{2}{3} u\).

In the next example, we’ll have variables and fractions on both sides of the equation.

Example \(\PageIndex{3}\):

Solve: \(x + \dfrac{1}{3} = \dfrac{1}{6} x − \dfrac{1}{2}\).

Exercise \(\PageIndex{5}\):

Solve: \(a + \dfrac{3}{4} = \dfrac{3}{8} a − \dfrac{1}{2}\).

Exercise \(\PageIndex{6}\):

Solve: \(c + \dfrac{3}{4} = \dfrac{1}{2} c − \dfrac{1}{4}\).

In Example 8.40, we’ll start by using the Distributive Property. This step will clear the fractions right away!

Example \(\PageIndex{4}\):

Solve: 1 = \(\dfrac{1}{2}\)(4x + 2).

Exercise \(\PageIndex{7}\):

Solve: −11 = \(\dfrac{1}{2}\)(6p + 2).

Exercise \(\PageIndex{8}\):

Solve: 8 = \(\dfrac{1}{3}\)(9q + 6).

Many times, there will still be fractions, even after distributing.

Example \(\PageIndex{5}\):

Solve: \(\dfrac{1}{2}\)(y − 5) = \(\dfrac{1}{4}\)(y − 1).

Exercise \(\PageIndex{9}\):

Solve: \(\dfrac{1}{5}\)(n + 3) = \(\dfrac{1}{4}\)(n + 2).

Exercise \(\PageIndex{10}\):

Solve: \(\dfrac{1}{2}\)(m − 3) = \(\dfrac{1}{4}\)(m − 7).

Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money and percent. But decimals are really another way to represent fractions. For example, 0.3 = \(\dfrac{3}{10}\) and 0.17 = \(\dfrac{17}{100}\). So, when we have an equation with decimals, we can use the same process we used to clear fractions—multiply both sides of the equation by the least common denominator.

Example \(\PageIndex{6}\):

Solve: 0.8x − 5 = 7.

The only decimal in the equation is 0.8. Since 0.8 = \(\dfrac{8}{10}\), the LCD is 10. We can multiply both sides by 10 to clear the decimal.

Exercise \(\PageIndex{11}\):

Solve: 0.6x − 1 = 11.

Exercise \(\PageIndex{12}\):

Solve: 1.2x − 3 = 9.

Example \(\PageIndex{7}\):

Solve: 0.06x + 0.02 = 0.25x − 1.5.

Look at the decimals and think of the equivalent fractions.

\[0.06 = \dfrac{6}{100}, \qquad 0.02 = \dfrac{2}{100}, \qquad 0.25 = \dfrac{25}{100}, \qquad 1.5 = 1 \dfrac{5}{10}\]

Notice, the LCD is 100. By multiplying by the LCD we will clear the decimals.

Exercise \(\PageIndex{13}\):

Solve: 0.14h + 0.12 = 0.35h − 2.4.

Exercise \(\PageIndex{14}\):

Solve: 0.65k − 0.1 = 0.4k − 0.35.

The next example uses an equation that is typical of the ones we will see in the money applications in the next chapter. Notice that we will distribute the decimal first before we clear all decimals in the equation.

Example \(\PageIndex{8}\):

Solve: 0.25x + 0.05(x + 3) = 2.85.

Exercise \(\PageIndex{15}\):

Solve: 0.25n + 0.05(n + 5) = 2.95.

Exercise \(\PageIndex{16}\):

Solve: 0.10d + 0.05(d − 5) = 2.15.

ACCESS ADDITIONAL ONLINE RESOURCES

Solve an Equation with Fractions with Variable Terms on Both Sides

Ex 1: Solve an Equation with Fractions with Variable Terms on Both Sides

Ex 2: Solve an Equation with Fractions with Variable Terms on Both Sides

Solving Multiple Step Equations Involving Decimals

Ex: Solve a Linear Equation With Decimals and Variables on Both Sides

Ex: Solve an Equation with Decimals and Parentheses

Practice Makes Perfect

In the following exercises, solve the equation by clearing the fractions.

• \(\dfrac{1}{4} x − \dfrac{1}{2} = − \dfrac{3}{4}\)
• \(\dfrac{3}{4} x − \dfrac{1}{2} = \dfrac{1}{4}\)
• \(\dfrac{5}{6} y − \dfrac{2}{3} = − \dfrac{3}{2}\)
• \(\dfrac{5}{6} y − \dfrac{1}{3} = − \dfrac{7}{6}\)
• \(\dfrac{1}{2} a + \dfrac{3}{8} = \dfrac{3}{4}\)
• \(\dfrac{5}{8} b + \dfrac{1}{2} = − \dfrac{3}{4}\)
• 2 = \(\dfrac{1}{3} x − \dfrac{1}{2} x + \dfrac{2}{3} x\)
• 2 = \(\dfrac{3}{5} x − \dfrac{1}{3} x + \dfrac{2}{5} x\)
• \(\dfrac{1}{4} m − \dfrac{4}{5} m + \dfrac{1}{2} m\) = −1
• \(\dfrac{5}{6} n − \dfrac{1}{4} n − \dfrac{1}{2} n\) = −2
• \(x + \dfrac{1}{2} = \dfrac{2}{3} x − \dfrac{1}{2}\)
• \(x + \dfrac{3}{4} = \dfrac{1}{2} x − \dfrac{5}{4}\)
• \(\dfrac{1}{3} w + \dfrac{5}{4} = w − \dfrac{1}{4}\)
• \(\dfrac{3}{2} z + \dfrac{1}{3} = z − \dfrac{2}{3}\)
• \(\dfrac{1}{2} x − \dfrac{1}{4} = \dfrac{1}{12} x + \dfrac{1}{6}\)
• \(\dfrac{1}{2} a − \dfrac{1}{4} = \dfrac{1}{6} a + \dfrac{1}{12}\)
• \(\dfrac{1}{3} b + \dfrac{1}{5} = \dfrac{2}{5} b − \dfrac{3}{5}\)
• \(\dfrac{1}{3} x + \dfrac{2}{5} = \dfrac{1}{5} x − \dfrac{2}{5}\)
• 1 = \(\dfrac{1}{6}\)(12x − 6)
• 1 = \(\dfrac{1}{5}\)(15x − 10)
• \(\dfrac{1}{4}\)(p − 7) = \(\dfrac{1}{3}\)(p + 5)
• \(\dfrac{1}{5}\)(q + 3) = \(\dfrac{1}{2}\)(q − 3)
• \(\dfrac{1}{2}\)(x + 4) = \(\dfrac{3}{4}\)
• \(\dfrac{1}{3}\)(x + 5) = \(\dfrac{5}{6}\)

In the following exercises, solve the equation by clearing the decimals.

• 0.6y + 3 = 9
• 0.4y − 4 = 2
• 3.6j − 2 = 5.2
• 2.1k + 3 = 7.2
• 0.4x + 0.6 = 0.5x − 1.2
• 0.7x + 0.4 = 0.6x + 2.4
• 0.23x + 1.47 = 0.37x − 1.05
• 0.48x + 1.56 = 0.58x − 0.64
• 0.9x − 1.25 = 0.75x + 1.75
• 1.2x − 0.91 = 0.8x + 2.29
• 0.05n + 0.10(n + 8) = 2.15
• 0.05n + 0.10(n + 7) = 3.55
• 0.10d + 0.25(d + 5) = 4.05
• 0.10d + 0.25(d + 7) = 5.25
• 0.05(q − 5) + 0.25q = 3.05
• 0.05(q − 8) + 0.25q = 4.10

Everyday Math

• Coins Taylor has $2.00 in dimes and pennies. The number of pennies is 2 more than the number of dimes. Solve the equation 0.10d + 0.01(d + 2) = 2 for d, the number of dimes.
• Stamps Travis bought $9.45 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 5 less than the number of 49-cent stamps. Solve the equation 0.49s + 0.21(s − 5) = 9.45 for s, to find the number of 49-cent stamps Travis bought.

Writing Exercises

• Explain how to find the least common denominator of \(\dfrac{3}{8}, \dfrac{1}{6}\), and \(\dfrac{2}{3}\).
• If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?
• If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?
• In the equation 0.35x + 2.1 = 3.85, what is the LCD? How do you know?

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

(b) Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/[email protected] ."

Texas Essential Knowledge and Skills (TEKS) 2nd Grade Math Skills

Printable Second Grade Math Worksheets, Study Guides and Vocabulary Sets.

Texas Essential Knowledge and Skills (TEKS) for Second Grade Math

Tx. 111.4. grade 2, adopted 2012., 2.2. number and operations. the student applies mathematical process standards to understand how to represent and compare whole numbers, the relative position and magnitude of whole numbers, and relationships within the numeration system related to place value. the student is expected to:, 2.2 (a) use concrete and pictorial models to compose and decompose numbers up to 1,200 in more than one way as a sum of so many thousands, hundreds, tens, and ones., 2.2 (b) use standard, word, and expanded forms to represent numbers up to 1,200., 2.2 (d) use place value to compare and order whole numbers up to 1,200 using comparative language, numbers, and symbols (&gt;, &lt;, or =)., 2.2 (e) locate the position of a given whole number on an open number line., 2.2 (f) name the whole number that corresponds to a specific point on a number line., 2.3. number and operations. the student applies mathematical process standards to recognize and represent fractional units and communicates how they are used to name parts of a whole. the student is expected to:, 2.3 (a) partition objects into equal parts and name the parts, including halves, fourths, and eighths, using words., 2.3 (b) explain that the more fractional parts used to make a whole, the smaller the part; and the fewer the fractional parts, the larger the part., 2.3 (d) identify examples and non-examples of halves, fourths, and eighths., 2.4. number and operations. the student applies mathematical process standards to develop and use strategies and methods for whole number computations in order to solve addition and subtraction problems with efficiency and accuracy. the student is expected to:, 2.4 (a) recall basic facts to add and subtract within 20 with automaticity., 2.4 (b) add up to four two-digit numbers and subtract two-digit numbers using mental strategies and algorithms based on knowledge of place value and properties of operations., 2.4 (c) solve one-step and multi-step word problems involving addition and subtraction within 1,000 using a variety of strategies based on place value, including algorithms., 2.5. number and operations. the student applies mathematical process standards to determine the value of coins in order to solve monetary transactions. the student is expected to:, 2.5 (a) determine the value of a collection of coins up to one dollar., 2.5 (b) use the cent symbol, dollar sign, and the decimal point to name the value of a collection of coins., 2.6. number and operations. the student applies mathematical process standards to connect repeated addition and subtraction to multiplication and division situations that involve equal groupings and shares. the student is expected to:, 2.6 (a) model, create, and describe contextual multiplication situations in which equivalent sets of concrete objects are joined., 2.6 (b) model, create, and describe contextual division situations in which a set of concrete objects is separated into equivalent sets., 2.7. algebraic reasoning. the student applies mathematical process standards to identify and apply number patterns within properties of numbers and operations in order to describe relationships. the student is expected to:, 2.7 (b) use an understanding of place value to determine the number that is 10 or 100 more or less than a given number up to 1,200., 2.7 (c) represent and solve addition and subtraction word problems where unknowns may be any one of the terms in the problem., 2.8. geometry and measurement. the student applies mathematical process standards to analyze attributes of two-dimensional shapes and three-dimensional solids to develop generalizations about their properties. the student is expected to:, 2.8 (b) classify and sort three-dimensional solids, including spheres, cones, cylinders, rectangular prisms (including cubes as special rectangular prisms), and triangular prisms, based on attributes using formal geometric language., 2.8 (c) classify and sort polygons with 12 or fewer sides according to attributes, including identifying the number of sides and number of vertices., 2.8 (d) compose two-dimensional shapes and three-dimensional solids with given properties or attributes., 2.9. geometry and measurement. the student applies mathematical process standards to select and use units to describe length, area, and time. the student is expected to:, 2.9 (c) represent whole numbers as distances from any given location on a number line., 2.9 (d) determine the length of an object to the nearest marked unit using rulers, yardsticks, meter sticks, or measuring tapes., 2.9 (g) read and write time to the nearest one-minute increment using analog and digital clocks and distinguish between a.m. and p.m., 2.10. data analysis. the student applies mathematical process standards to organize data to make it useful for interpreting information and solving problems. the student is expected to:, 2.10 (a) explain that the length of a bar in a bar graph or the number of pictures in a pictograph represents the number of data points for a given category., 2.10 (c) write and solve one-step word problems involving addition or subtraction using data represented within pictographs and bar graphs with intervals of one., 2.10 (d) draw conclusions and make predictions from information in a graph., newpath learning resources are fully aligned to us education standards. select a standard below to view correlations to your selected resource:.

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Converting Fractions to Decimals Lesson Plan: Numbers and Operations– Fractions

*Click to open and customize your own copy of the  Converting Fractions to Decimals Lesson Plan . 

This lesson accompanies the BrainPOP topic Converting Fractions to Decimals , and supports the standard of using decimal notation for fractions with denominators 10 or 100. Students demonstrate understanding through a variety of projects.

Step 1: ACTIVATE PRIOR KNOWLEDGE

Ask students:

• Where do you see decimals in everyday life? Where do you see fractions?
• How are decimals and fractions alike? 

Step 2: BUILD KNOWLEDGE

• Read the description on the Converting Fractions to Decimals topic page .
• Play the Movie , pausing to check for understanding. 
• Have students read one of the following Related Reading articles: “In Practice,” or “Personalities.” Partner them with someone who read a different article to share what they learned with each other.

Step 3: APPLY and ASSESS 

Students take the Converting Fractions to Decimals Quiz , applying essential literacy skills while demonstrating what they learned about this topic.

Step 4: DEEPEN and EXTEND

Students express what they learned about converting fractions to decimals while practicing essential literacy skills with one or more of the following activities. Differentiate by assigning ones that meet individual student needs.

• Make-a-Movie : Produce a news report about a race where athlete A ran .2 of the race in five minutes and athlete B 75/100 of the race in that time. Explain who is in the lead, and how you know. 
• Make-a-Map : Identify the relationship between fractions and decimals. Outline the steps to convert between the two. 
• Creative Coding : Code a math problem that involves converting a fraction into a decimal, and challenge a classmate to solve. 

More to Explore 

Battleship Numberline : Players estimate paper ships’ positions on a number line in this interactive game. 

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• Pause Point Overview : Video tutorial showing how Pause Points actively engage students to stop, think, and express ideas.  
• Learning Activities Modifications : Strategies to meet ELL and other instructional and student needs.
• Learning Activities Support : Resources for best practices using BrainPOP.

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Course: 7th grade   >   Unit 2

• Solving percent problems
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Video transcript

What is 2.10 as a fraction?

Numbers can be represented in a multitude of ways, from decimals, to percentages to fractions. In this article, we will demonstrate how you can convert a decimal number to a fraction and soon you’ll be a pro at it, too!

Solution: 2.10 as a fraction is 21/10

Converting 2.10 to a fraction, step-by-step.

The first step to converting 2.10 to a fraction is to re-write 2.10 in the form p/q where p and q are both positive integers. To start with, 2.10 can be written as simply 2.10/1 to technically be written as a fraction.

Next, we will count the number of fractional digits after the decimal point in 2.10, which in this case is 2. For however many digits after the decimal point there are, we will multiply the numerator and denominator of 2.10/1 each by 10 to the power of that many digits. For instance, for 0.45, there are 2 fractional digits so we would multiply by 100; or for 0.324, since there are 3 fractional digits, we would multiply by 1000. So, in this case, we will multiply the numerator and denominator of 2.10/1 each by 100:

2.10 × 100 1 × 100 = 210 100 \frac{2.10 × 100}{1 × 100} = \frac{210}{100} 1 × 100 2.10 × 100 ​ = 100 210 ​

Now the last step is to simplify the fraction (if possible) by finding similar factors and cancelling them out, which leads to the following answer:

210 100 = 21 10 \frac{210}{100} = \frac{21}{10} 100 210 ​ = 10 21 ​

Convert Other Values to Fractions

Become a pro at converting decimals or percentages to fractions by exploring some examples, like the ones below:

What is 85.19 as a fraction?

What is 37.8 as a fraction?

What is 4.086 as a fraction?

What is 516.7 as a fraction?

What is 15.52 as a fraction?

Download FREE Math Resources

Take advantage of our free downloadable resources and study materials for at-home learning.

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